GEOMETRY OF LINK INVARIANTS
By
SERGEY A. MELIKHOV
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
ACKNOWLEDGMENTS
I am indebted to Prof. A. N. Dranishnikov for his encouragement and guid
ance through my study in the graduate school, to Prof. D. Cenzer, Prof. J. Keesling,
Prof. S. Obukhov and Prof. Yu. B. Rudyak for their interest in my dissertation, and to the
faculty of the Mathematics Department for their hospitality.
Last, but not least, I would like to thank Dr. P. M. Akhmetiev for stimulating
correspondence and discussions and Juan Liu for her support and patience.
TABLE OF CONTENTS
page
ACKNOW LEDGMENTS ................................. ii
ABSTRACT . . . . . . . . . . iv
CHAPTER
1 INTRODUCTION .................................. 1
2 COLORED FINITE TYPE INVARIANTS AND kQUASIISOTOPY ...... 3
3 THE CONWAY POLYNOMIAL .......... ................ 10
4 THE MULTIVARIABLE ALEXANDER POLYNOMIAL ........... .18
5 THE HOMFLY AND KAUFFMAN POLYNOMIALS ............. 32
REFERENCES ................................. .... 34
BIOGRAPHICAL SKETCH ..................... .......... 37
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
GEOMETRY OF LINK INVARIANTS
By
Sergey A. Melikhov
August 2005
C'!I ii: Alexander N. Dranishnikov
A, i ,r Department: Mathematics
kQuasiisotopy is an equivalence relation on piecewiselinear (or smooth) links in the
Euclidean 3space, whose definition closely resembles the first k stages of the construction
of a Casson handle. It generalizes and refines both kcobordism of Cochran and Orr and
klinking of S. Eilenberg, as modified by N. Smythe and K. Kc.1 li ili kQuasiisotopy
turned out to be helpful for understanding the geometry of the Alexander polynomial and
Milnor's pinvariants, leading, in particular, to the following results.
There exists a direct multivariable analogue of the Conway polynomial VL(z),
i.e., a (finite) polynomial VL(z,,... Zn) with integer coefficients, satisfying L 
VL = ZiVLo for any Conway triple (L+, L_, Lo) whose disagreement is within the
components of the ith color. Moreover, for 2component links all coefficients of the
2variable VL are integer lifting of Milnor's invariants p(1 ... 12... 2) of even length.
Each Cochran's derived invariant /3 (defined originally when the linking number
vanishes) extends to a Vassiliev invariant of 2component links.
If Ki are the components of a link L, each coefficient of the power series VL/ H V7K
is invariant under sufficiently close Coapproximation, and can be (uniquely)
extended, preserving this property, to all topological links in R3. The same holds for
the power series VLI(z,..., z,)/ H VKj(z), as well as HL/ HK, and FL/H FKi,
where HL and FL are certain exponential parameterizations of the twovariable
HOMFLY and Kauffman polynomials.
* No difference between PL isotopy and TOP isotopy (as equivalence relations on PL
links in R3) can be detected by Vassiliev invariants.
CHAPTER 1
INTRODUCTION
Recall that PL isote'ip; of links in S3 can be viewed as the equivalence relation
generated by ambient isotopy and insertion of local knots. (Thus any knot in S3 is PL
isotopic to the unknot.) The "links modulo knots" version of the Vassiliev Conjecture for
knots [37] could be as follows.
Problem 1.1. Are PL isote'i~; classes of links in S3 separated by finite type invariants
that are well,,. ,,. .1 up to PL isote'i,?
It is to be noted that both questions depend heavily on the topology of S3 (or R3)
as opposed to other 3manifolds. Indeed, the answer to 1.1 is negative for links (in fact,
even knots) in any contractible open 3manifold W other than R3 ([25] and Theorem
2.2 below), whereas the original Vassiliev Conjecture fails for knots in any such W
embeddable in R3 (e.g., in the original Whitehead manifold) [7, 8]. Similarly, if M is a
homotopy sphere, then M S3 given either a positive solution to 1.1 for knots in M ([25]
and Theorem 2.2 below), or a proof of the Vassiliev Conjecture for knots in M#M [7, 8].
Some further geometry behind Problem 1.1 is revealed by
Theorem 1.2. The answer to 1.1 is affirmative if and only if the following four state
ments (on PL links in S3) hold simultii, ..ul;,
(i) finite type invariants, wellI, ., ,..1 up to PL isote'i,; are not weaker than finite
KLtype invariants (cf. 2), well,. .,, .1 up to PL isote'i~
(ii) i .i., /li,i,.i]il /lii by finite KL'i:,,,. invariants, well,., i,, ,l up to PL isotei,',
implies kquasiisotei'; for all k E N;
(iii) kquasiisotclji, for all k E N implies TOP ':,'/,,;;, in the sense of Milnor (i.e.,
homote'i.; within the class of to'', 1 I. ':. l embeddings);
(iv) TOP isot,'pi, implies PL '. 4.'./,/
The assertion follows since the converse to each of (i)(iv) holds with no proviso: to
(ii) by Theorem 2.2 below, to (iii) by Theorem 1.3(b) below, to (i) and (iv) by definitions.
We recall that PL links L, L': mS1 S3 (where mS1 = S' U ... U S'J are called
kquasiisotopic [25], if they are joined by a generic PL homotopy, all whose singular
links are kquasiembeddings. A PL map f: mS1 > S3 with precisely one double
point f(p) = f(q) is called a kquasiembedding, k E N, if, in addition to the singleton
Po := {f(p)}, there exist compact subpolyhedra Pi,..., Pk C S3 and arcs Jo,... Jk C mS'
such that f'(Pj) C Jj for each j k and Pj U f(Jj) C Pj+i for each j < k, where the
latter inclusion is nullhomotopic for each j < k.
This definition is discussed in detail and illustrated by examples in Melikhov and
Repos [25] and [26]. The only result we need from these papers is the following obvious
Theorem 1.3. [25] (a) For each k E N, kquasiisote'i', classes of all suff. ill, close PL
approximations to a t('l'. 1..':. 'l link coincide.
(b) TOP isotopic PL links are kquasiisotopic for all k E N.
The paper is organized as follows. 2 contains a little bit of general geometric theory
of finite KLtype invariants, which we interpret in the context of colored links. (We do
not attempt at extending any of the standard algebraic results of the theory of finite type
invariants.) The rest of the paper is devoted to specific examples. Except for invariants
of genuine finite type from the Conway polynomial (3), the multivariable Alexander
polynomial (Theorems 4.2 and 4.3) and the HOMFLY and Kauffman polynomials (5),
we need to work just to produce interesting examples (Theorems 4.4 and 4.6). The
p ,off is a further clarification of the relationship between the Alexander polynomial and
geometrically more transparent invariants of Milnor [28] and Cochran [5] (Theorem 4.8).
We are also able to extract additional geometric information (Theorem 3.4 and Corollary
4.10b), which was the initial motivation for this paper (compare Problem 1.5 in Melikhov
and Repovs [25]).
CHAPTER 2
COLORED FINITE TYPE INVARIANTS AND kQUASIISOTOPY
We introduce finite type invariants of colored links, as a straightforward generalization
of the KirkLivingston setting of finite type invariants of links [16]. KirkLivingston
type k invariants are recovered as type k invariants of colored links whose components
have pairwise distinct colors, whereas the usual type k invariants of links (as defined by
Goussarov [9] and Stanford [33]) coincide with type k invariants of monochromatic links.
Although normally everything boils down to one of these two extreme cases, the general
setting, apart from its unifying appeal, may be useful in proofs (cf. proof of Proposition
3.1) and in dealing with colored link polynomials (cf. Theorem 4.6). Refer to the following
bracketed citations for a background on finite type invariants: [37, 10, 31, 4, 33, 3].
An mcomponent classical colored link map corresponding to a given coloring
c: {1,...,m} {1,...,m} is a continuous map L: mS1  S3 such that L(S/')Ln (S) = 0
whenever c(i) / c(j). Thus usual link maps correspond to the identity coloring and ar
bitrary maps to the constant coloring. Let LM,3...,1(c) be the space of all PL clink maps
mS1  S3, and let M(c) denote its subspace consisting of maps whose only singularities
are transversal double points (the integer m will be fixed throughout this section). Note
that M(c) is disconnected unless c is constant. Let LIM (c) (resp. LM>,(c)) denote the
subspace of LM(c) consisting of maps with precisely (resp. at least) n singularities. Note
that L./i,(c) does not depend on c.
Given any ambient isotopy invariant X : C I,,(c)  G with values in a finitely
generated abelian group G, it can be extended to LM(c) inductively by the formula
X(L,) X(L+) X(L),
where L+, L_ e L (c) differ by a single positive crossing of components of the same
color, and L E C 1 ,I (c) denotes the intermediate singular link with the additional
double point. If thus obtained extension X vanishes on LM>r+i(c), i.e., on all clink maps
with at least r + 1 transversal double points, for some finite r, then X is called a finite
ctype invariant, namely of ctype r. Note that in our notation any ctype r invariant is
also of ctype r + 1. Sometimes we will identify a finite ctype invariant X: L/, (c)  G
with its extension X: LM(c)  G.
For m 1 there is only one coloring c, and finite ctype invariants, normalized by
x(unknot) = 0, coincide with the usual GoussarovVassiliev invariants of knots. If c is a
constant map, we write "type" instead of "ctype", and if c is the identity, we write "KL
type" instead of "ctype", and LM, instead of LM (c). If the coloring c is a composition
c = de, then any dtype k invariant is also a ctype k invariant. In particular, any type k
invariant is a KLtype k invariant, but not vice versa. Indeed, the linking number lk(L)
and the generalized SatoLevine invariant (see 3) are of KLtypes 0 and 1, respectively,
but of types exactly 1 and 3.
It is arguable that triple and higher pinvariants with distinct indices should be
regarded as having KLtype 0 (as they assume values in different cyclic groups depending
on link homotopy classes of proper sublinks, we must either agree to imagine these groups
as subgroups, iv, of Z E Q/Z, or extend the definition of finite KLtype invariants to
include the possibility where each pathconnected component of LM maps to its own
range). In contrast, there are almost no finite type invariants of link homotopy [24],
although this can be remedied by consideration of string links (see references in Hughes
[12] or Masbaum and Vaintrob [23]) or partially defined finite type invariants [9, 12]. We
show in 4 that certain pinvariants of 2component links have integer lifting of finite
KLtype, at least some of which are not of finite type due to
Proposition 2.1. The KLtype 0 invariant v(L) = (l)lk(L) is not offinite type.
Proof. By induction, v(Ls) = 2k for any singular link L8 with k transversal intersections
between distinct components and no selfintersections. O
It is clear that KLtype r invariants X : C  G form an abelian group, which we
denote by G, (or G' ) when the number of components needs to be specified). Clearly, Go
is the direct sum of 7Io(CM)l copies of G. Let G,(A) denote the subgroup of Gr consisting
of invariants vanishing on all links whose link homotopy class is not AE 7 o(CM), and
on a fixed link B\ E A with unknotted components. The latter can alv, be achieved
by adding a KLtype 0 invariant, so the quotient map q: Gr  G,/Go takes G,(A)
isomorphically onto a subgroup, independent of the choice of B%; moreover G,/Go
%x q(G,(A)). For any 1k E Z, it was proved in Kirk and Livingston [16] that Z 2)(lk) Z
(generated by the generalized SatoLevine invariant) and conjectured that Z 2)(1k) is
not finitely generated for r > 1. It is wellknown [33] that the group of monochromatic
Gvalued type r invariants of mcomponent links is finitely generated; in particular, so
is Gr Let Gr denote the subgroup of Gr consisting of those invariants which remain
unchanged under tying local knots, i.e., under PL isotopy. Notice that the wellknown
proof that type 1 invariants of knots are trivial works also to show that G1 = G1. In the
following remarks we address the difference between Gr and Gr for r > 1.
Remarks. (i). By evaluating any type r invariant of knots on the components of a link
L E A we obtain a monomorphism
A: eC)G GCr(A).
i= 1
We claim that its image (whose elements we regard here as the "least interesting among
all) is a direct summand, with complement containing Gr(A). Indeed, for any X E Gr(A)
define a knot invariant Xi by i(K) x= (Ki), where K, is the link, obtained by tying
the knot K locally on the ith component of B\. Then Xi is a type r knot invariant, since
the local knot on the ith component of Ki, viewed as a knotted ball pair (B3, B1), can
be chosen to look precisely the same as K outside a small ball. Define an endomorphism
4) of Gr(A) by Q(X)(L) X(L) EC~ L (i ), where Lt denotes the ith component of
L. Then 4) takes any X E imA to zero and any X E Gr(A) to itself, and consequently
defines a splitting (depending, in general, on the choice of B\) of the quotient map
G,(A) cokerA, with image containing G,(A).
(ii). We claim that an invariant X in the complement to imA is invariant under tying local
knots iff the restriction of X to 1M1 (which is a KLtype r 1 invariant of singular links)
is. Indeed, the "only if" implication is trivial. Conversely, it suffices to find one link L E A
such that X(L) is unchanged when any local knot is added to L. Since X = t(X), clearly
B\ is such a link.
In particular, since KLtype 1 invariants of singular links are invariant under tying
local knots due to the oneterm (framing independence) relation, it follows that
G2(A) G2(A) imA.
Theorem 2.2. Let X be a KLtype k invariant. If x is invariant under PL isotei,;, then
it is invariant under kquasiisot(i',1
f. For k = 0 there is nothing to prove, so assume k > 1. It suffices to show that any
x E Gk vanishes on any kquasiembedding f: mS1 S3 with precisely one double point
f(p) f(q). Let Po,..., Pk and Jo,..., Jk1 be as in the definition of kquasiembedding,
and let Jo denote the subarc of Jo with 9Jo = {p, q}. In order to have enough room
for general position we assume that Pi's are compact 3manifolds (with boundary) and
Jo C Int Jo (this can be achieved by taking small regular neighborhoods).
Since the inclusion Po U f(Jo) c P1 is nullhomotopic, there exists a generic PL
homotopy fJ: f(mS1) S S3 such that the homotopy ft:= f"f: mS1 S3 satisfies fo =f,
ft(Jo) C Pi, ft = f outside Jo, and f1(Jo) is a small circle, bounding an embedded disk in
the complement to fi(mS1). Using the oneterm (framing independence) relation, we see
that any finite KLtype invariant vanishes on fl. Hence any KLtype 1 vanishes on fo = f,
which completes the proof for k = 1. O
Before proceeding to the general case, we state the following generalization of the
oneterm (framing independence) relation.
Lemma 2.3. If f C 1 has all its double points inside a ball B such that f (B) is an
arc, then for ,1:,1 x E G,, x(f) = 0.
F *' By induction on n. O
Proof of 2.2 continuedd). Let us study the jump of our invariant X E Gk on the link maps
ft occurring at singular moments of the homotopy f[. Let ft, E C 11 be one. Using
the definition of kquasiisotopy, k > 2, we will now construct a generic PL homotopy
ftl,t: ft,(mS1) S3 such that the homotopy ft,,t := f',tf mS S3 satisfies ft,o = fti,
fti,t(J1) C P2, fti,t = ft outside Ji, and fti,i takes J1 into a ball B1 in the complement to
ft,,i(nS1 \ Ji).
Indeed, since ft (Ji) C P1 U f(J1) is nullhomotopic in P2, we can contract the
1dimensional polyhedron ft, (J1 \ U), where U is a regular neighborhood of &JI in Ji,
embedded by ft,, into a small ball B' C S3 \ ft,(mS1 \ Ji), by a homotopy ft,,tl j as
required above. Joining the endpoints of ft,(mS1 \ J1i) to those of fti,t(Ji \ U) by two
embedded arcs ft,,t(U) in an arbitrary way (continuously depending on t), we obtain the
required homotopy. Taking a small regular neighborhood of B, U fti,,(J1) relative to
ft1,l(aJi) we obtain the required ball B1.1
By Lemma 2.3, any finite KLtype invariant vanishes on ft,il. The homotopy ft,,t as
well as the analogously constructed ft/,t for each ft, c C11 does not change any invariant
of KLtype 2, so for k = 2 we are done. The proof in the general case must be transparent
now. For completeness, we state E
2.4. The nth inductive step. For each critical level ft, ....,, E C V i of I,,:. of
the homotopies constructed in the previous step, there exists a generic PL homote'l':
fti,(....t,t i ft;s .... t (nS1) S3 such that the homoter'i fti .....tj,t .= fl^....t,,,ttft,....ti,
(i) ft ,....,tt = ft, ....t, for t = 0 and outside J,;
(ii) ftj,... t,t(j,) C Pn+1 for each t;
(ii) f ...t (B ) J= for some PL 3ball Bn1. O
Remark. In fact, the proof of Theorem 2.2 works under a weaker assumption and with a
stronger conclusion. Namely, kquasiisotopy can be replaced with virtual kquasiisotopy
(see definition in Melikhov and Repovs [25]), whereas indistinguishability by KLtype k
1 Using the notation from Melikhov and Repovs [25], the homotopy ft,,t can be visual
ized as shifting the arcs f(I) onto the arcs F(Ij) and then taking the image of J1 into a
ball along the track of the nullhomotopy F.
invariants (which can be thought of as a formal generalization of kequivalence in the sense
of Goussarov [10], Habiro and Stanford) can be replaced with geometric kequivalence,
defined as follows. For each n > 0, let L.1 ,, denote the subspace of C11 consisting of
the link maps I such that all singularities of I are contained in a ball B such that 1 (B) is
an arc. We call link maps 1,1' E C11 geome l,.:. a'll kequivalent if they are homotopic in
the space C 1 U C 1, i I;k, where C 11, for k > 0, i > 0 consists of those link maps with i
singularities which are geometrically (k 1)equivalent to a link map in C 1 ,,
To see why geometric kequivalence may differ from kquasiisotopy, we employ the
notion of weak kquasiisotopy, which is defined similarly to kquasiisotopy, but with
nullhomotopies replaced by nullhomologies, cf. [26].
The difference between geometric kequivalence and weak kquasiisotopy is analogous
to the difference between the lower central series and the derived series, and for k > 2
can be visualized as follows. Let i 1, denote the kth Milnor link (cf., e.g., Fig. 1 in
Melikhov and Repovs [26]), and let M7A k > 2, be 11, where the "1.,i,; component
is replaced by its Whitehead double. Unclasping the clasp of the doubled component
is a 1quasiisotopy ht from 1W to the unlink. This ht is not a 2quasiisotopy (even
not a weak 2quasiisotopy), since i i, is not a boundary link, as detected by Cochran's
invariants (cf., e.g., [26]). It would thus be in the spirit of Conjecture 1.1 in Melikhov and
Repovs [26] to conjecture that there exists no 2quasiisotopy, even weak, from M3 to the
unlink. However, since il, is (k 1)quasiisotopic to the unlink, ht realizes geometric
kequivalence between MW and the unlink.
For k = 1 no such example exists, since geometric 1equivalence clearly implies weak
1quasiisotopy. For 2component links the converse also holds (cf. Fig. 2 in Milnor [27]),
amplifying the analogy with the lower central series and the derived series, whose respec
tive initial terms coincide. Thus the difference between 1quasiisotopy and geometric
1equivalence for 2component links reduces to the difference between 1quasiisotopy and
weak 1quasiisotopy (see Fig. 2(d) in Melikhov and Repovs [25]).
Existence of difference between geometric kequivalence and "indistinguishability by
KLtype k invariants, welldefined up to PL isotopy," already for k = 1, would follow
9
from the (expected) positive solution to Problem 1.5 in Melikhov and Repovs [25] for weak
1quasiisotopy.
CHAPTER 3
THE CONWAY POLYNOMIAL
Recall that the Co,', '1.;/ / ;;,,I I.:Il/ of a link L is the unique polynomial VL(z)
satisfying unknot = 1 and the crossing change formula
VL () vL (z) = ZV (), (1)
where L+ and L_ differ by a positive crossing change, and Lo is obtained by oriented
smoothing of the selfintersection of the intermediate singular link L,. Note that Lo
has one more (respectively less) component than L+ and L_ if the intersection is a
selfintersection of some component (resp. an intersection between distinct components).
The skein relation (1) shows that the coefficient at zk in VL is a type k invariant. The
generalized oneterm relation for finite type invariants specializes here to the equation
L = 0 for any split link L (i.e., a link whose components can be split in two nonempty
parts by an embedded sphere). Using (1) it is now easy to see that the Conway polyno
mial of an mcomponent link L is necessarily of the form
VL(z) z= zm(co + C1z2 + C+z2n)
for some n. By (1), co(K) = 1 for any knot K, which can be used to recursively evaluate
co on any link L. For example, it is immediate that co(L) is the linking number of L for
m = 2, and co(L) = ab + be + ca for m = 3, where a, b, c are the linking numbers of the 2
component sublinks (cf. [22]). For arbitrary m, it is easy to see that co(L) is a symmetric
polynomial in the pairwise linking numbers (and thus a KLtype 0 invariant); see [18, 23]
for an explicit formula.
Proposition 3.1. ck(L) is of KL;1,1,' 2k.
Proof. Let L be a ccolored link, and let Icl denote the number of colors used, i.e., the
cardinality of the image of c. Then the coefficient of VL at z1 l1 is either co(L) or 0
according as c is onto or not; in either case, it is a ctype 0 invariant. An induction on i
using the skein relation (1) shows that the coefficient of 7L at z 1l1+i is of ctype i. In the
case where c is the identity, this is our assertion. E
In order to compute VK for a knot K, we could consider a sequence of crossing
changes in a plane diagram of K, turning K into the unknot (cf. [31]). Then VK
Vunknot + z L CiL,, where Li are the 2component links obtained by smoothing the
crossings and Ei = 1 are the signs of the crossing changes. We could further consider a
sequence of crossing changes in the diagram of each Li, involving only crossings of distinct
components and turning Li into a split link. This yields L, = z xCEij~Ky, where Kij are
the knots obtained by smoothing the crossings and ij = 1 are the signs of the crossing
changes. Since the diagram of each Kij has fewer crossings than that of K, we can express
VK, iterating this procedure, as Yk z2k(y Cki Vunknot), where the signs Cki = 1 are
determined by the above construction. Since unknot = 1, plainly VK = k z 2k i eki
Note that this procedure shows that VK is indeed a polynomial (rather than a power
series) for any knot K; a similar argument works for links.
Now let L be a link, and suppose that L' is obtained from L by tying a knot K
locally on one of the components. We can echo the above construction, expressing VL' as
>Ck z2k i ki VL, where the signs eki are same as above. Thus' VL = LVK. It follows
that the power series
7 IN VL( )
S K, (2) Z VKm (Z)
where K1,..., Km are the components of L, is invariant under PL isotopy of L. Note
that the above formula can be rewritten as V17 VL (VKI ". VK, 1)V1, mean
while VK, "" Km is of the form 1 + b1z2 + . + bz2" for some n, where bi(L) =
i+...+i ci(K1) .. cim (K,). We find that V1 is of the form
V/(z) z (ao + aZ2 + a2 +...),
1 The conclusion is, of course, wellknown, cf. [20], but we need the argument in order to
set up notation for use in the proof of Theorem 3.4.
where a, = c (a_ilbl + + (,,1' ). Hence V1 c Z[[z]] (rather than just Q[[z'l]]), and
Ck ck (mod gcd(co,... Ck1)). In particular, for m = 2 we see that co(L) = co(L) is the
linking number, and ac(L) = cl(L) co(L)(cl(Ki) + cl(K2)) is (cf. [22]) the generalized
SatoLevine invariant!
Under the generalized SatoLevine invariant we mean the invariant that emerged
in the work of PolyakViro (see [2]), KirkLivingston [16, 22], Akhmetiev (see [1]) and
N I.: ,iiiOhyama [30]; see also [25].
Remark. For m = 2 we can also obtain a (L) from cl(L) by applying the projection
+ from Remark (i) in 2, with a certain choice of B\ (cf. [16]). This is not surprising,
because Cl has KLtype 2 by Proposition 3.1, hence N(cl) is invariant under PLisotopy by
Remark (ii) in 2.
Remark. Clearly, the power series V7 is actually a polynomial if the components of L are
unknotted or, more generally, have no nonlocal knots. Due to a splitting of the multi
variable Alexander polynomial (see 4), V7 splits into a product of a polynomial (which is
a quotient of the original VL) and a power series, both invariant under PL isotopy.
Theorem 3.2. (i) For each L: mS1 c S3 and 1,':1 n E N there exists an F, > 0 such that
if L': mS1 S3 is CO Eclose to L,
V,(z) 7V(z) mod (z").
(ii) V1 can be ;,,.:,;,;. 1;, extended to all tol'. I1.:. 'l links in S3, preserving (i).
(iii) The extended V1 is invariant under TOP isote'i;, of L.
Of course, the extended V* need not be a rational power series for some wild links
(which therefore will not be TOP isotopic to any PL links).
Proof. The coefficient dk of V1 at zk is of (monochromatic) type k since it is a polynomial
in the coefficients of the Conway polynomials of L and its components, homogeneous
of degree k with respect to the degrees (in z) of the corresponding terms. Since dk is
invariant under PL isotopy, by Theorem 2.2 it is invariant under kquasiisotopy. The
assertions (i) and (ii) now follow from Theorem 1.3(a), and (iii) either from [25] or,
alternatively, from (ii) and compactness of the unit interval. E
Proposition 3.1 implies that each ak(L) is of KLtype 2k. It is easy to check that
aI(L) = cl(L) co(L) (i cl(Ki)) is, in fact, of KLtype 1 and that
a2(L) 2(L) (L) (( c(KL))) co(L) c2(K ) + Yc + c(Ki)c1(K)
is of KLtype 3.
Proposition 3.3. a2(L) is not of KLtype 2 form = 2.
Proof. Consider the case lk(L) = 0, then 2(L) = c2(L) cl(L)(ci(Ki) + ci(K2)), which
can be also written as as(L) a3(L)(a2(Ki) + a2(K2)), where ai(L) denotes the coefficient
of VL at z'. The "third d. i . ,i v., of ci2(L), i.e., restriction of a2(L) to M3 can be found
using the Leibniz rule (XV)(L,) = x(L+)(Ls) + X(L,) (L_). In the case where the 3
singular points are all on the same component Ksss, it is given by
a2(Lsss) = a2(Looo) a2(L++o)ao(Koo) a2(+o+)ao(Koo) a2(Lo++)ao(Koo)
where, as usual, + and stand for the overpass and the underpass, and 0 for the smooth
ing of the crossing s. Assuming that each L** on the right hand side has three compo
nents and each K,, only one component, we can simplify this as co(Looo) co(L++o) 
co(Lo+o) co(Loo+).
Let o: S1 c S2 be a generic C1approximation with 3 double points of the clockwise
double cover S1 S1 C R2, and let A, B, C, D denote the 4 bounded components of
R2 \ c(S1) such that D contains the origin. Let Ks, e M31 be the composition of p and
the inclusion R2 C S2 c S3, and let K: S1 S3\K.ss(S1) be a knot in the complement of
Kss linking the clockwise oriented boundaries of A, B, C, D with linking numbers a, b, c, d
such that a + b + c + 2d= 0. Finally, define Lss, E LM2 to be the union of K and Kss,
then Ik(L.ss) a + b + c + 2d 0 and we find that
a2(Lss) = (a + b + c+ d)d (a + c+ d)(b + d) (a + b + d)(c + d) (b + c+ d)(a + d).
14
This expression is nonzero, e.g., for a = 1, b = 2, c = 3, d = 0. E
By Proposition 3.1 and Theorem 2.2, a,(L) is invariant under 2nquasiisotopy.
However, according to Proposition 3.3, the following strengthening of this assertion cannot
be obtained by means of Theorem 2.2.
Theorem 3.4. an(L) is invariant under nquasiisote'ii
The proof makes use of the following notion. We define colored link homote('i':
to be the equivalence relation on the set of links colored with m colors, generated by
intersections between components of the same color (including selfintersections) and
addition of trivial component of any color, separated from the link by an embedded
sphere. Thus an mcomponent colored link L is colored link homotopic with an (m + k)
component colored link L' iff L' is homotopic to L U Tk through colored link maps, where
Tk is the kcomponent unlink, split from L by an embedded sphere and colored in some
way. Such a homotopy will be called a colored link homotopy between L and L'.
P . Let us start by considering the above procedure for computing the Conway
polynomial of a knot K in more detail. One step of this procedure yields Ck(K) =
ck(unknot) + E Cick(Li) and ck(Li) = EijCk l(Kyi). Since co = 1 for every knot, Ck(K)
can be computed in k steps, regardless of the number of crossings in the diagram of K.
Moreover, if one is only interested in finding Ck(K) for a given k (which would not allow,
e.g., to conclude that VK is a polynomial not just a power series), the computation could
be based on arbitrary generic PL homotopies rather than those iri. I 1 by the diagram
of K. In particular, we allow (using that co = 0 for every link with > 1 components and
each ci = 0 for a trivial link with > 1 components) selfintersections of components in the
homotopies from Li's to split links, so that Kij's may have three and, inductively, any odd
number of components. For such an n step procedure, the equality VK = k,, z2k 1 i kiT
where the signs eki are determined by the homotopies, holds up to degree 2n.
Now let L be an mcomponent link, and suppose that L' is obtained from L by a
generic PL link homotopy H with a single singular level L,. Color the components of L
with distinct colors, then the (m + 1)component smoothing Lo of Ls is naturally colored
with m colors. Suppose inductively that Loi2...i, is one of the smoothed singular links
in a generic PL colored link homotopy Ho0i...i between the link Loi2...i and some link
L', which either has one less component than L,, or coincides (geometrically) with
L. If there exist such homotopies Hoi2...i0 for 1 = 1,...,2k, we iv that L8 is a +()
skein kquasiembedding, provided that the selfintersection in H is positive (negative).
A skein kquasiisote'l' is a link homotopy where every singular level is a + or skein
kquasiembedding, depending on the sign of the selfintersection.
To see that a, is invariant under skein nquasiisotopy 2 we return to the orig
inal link homotopy H and denote by K and K' the components of L and L' each
concolor with two components of Lo. As in the above argument for knots, we have
VK = Ekn Z2k(iEki ) + z2n+1P and z L kn 2k i ki1m L) + z2n+lQ,
where P(z) and Q(z) are some polynomials, and the signs Cki are determined by the
homotopies Hoi2... Then
ZImVL z1 LR + z2n+lp P zI VL 2n+
VK' VKR + z2n+lQ VK
where R(z) = YEkn z2k Zi 6ki and S(z) is some power series, and the assertion follows.
To complete the proof, we show that nquasiisotopy implies skein nquasiisotopy.
It suffices to consider a crossing change on a component K of L, satisfying the definition
of nquasiisotopy. Let f, Jo, ... iJ and Po,... Pn be as in the definition of nquasi
embedding; we can assume that P1 contains a regular neighborhood of f(Jo) containing
L(Jo). We associate to every link Lo,, such that L \ K is a geometric sublink of Lo,, but
L itself is not, the collection of positive integers d(Lo) = (do, ... d,), where di for i > 0
(resp. i = 0) is the minimal number such that the ith component of Lo, not in L \ K is
nullhomotopic in Pd, (resp. is homotopic to K with support in Pd). It is easy to see that
2 By the proof of Theorem 4.9 below, and since ca,2n and a2n,0 in Theorem 4.8 depend
only on the linking number, for n > 0 each a, is actually invariant under skein (n )
quasiisotopy.
d(Loi2...i +) is obtained from d(Loi ...i) by one of the following two operations:
(do,...,di,...,dm) (do,... ,di + ,di + ,... ,d );
(do,..., d ,..., dj,..., dm) (do,..., m ax(d d ),. d ,. dm).
Conversely, if for some i the operation ai and all operations 3ij 's lead to collections of
integers not exceeding n, one can construct a homotopy Ho0i...i between Loi2...i and some
L", which either has one less component than L,, or coincides with L.
Suppose that none of the integers d(Loi2...i,) exceeds 1 + 1, and at least r of them do
not exceed 1. If r > 1, let di be one of these r, then none of the integers 7(d(Loi ...i )),
where 7 is ai or /,i, exceeds 1 + 1, and at least r 1 of them do not exceed 1. Thus if at
least two components of Loi2...i are not in L\K and none of the integers d(Loi2...i) exceeds
1, we can construct a homotopy Ho0i...i as above, and for each singular link Loi2...ik+ in
this homotopy also a homotopy Hoi2,...i as above, so that for each singular link L,, 2
in this homotopy, none of the integers d(Loi2...i+2) exceeds 1 + 1. But it is indeed the
case for k = 1, hence for any odd k, that at least two components of Loi2...ik are not in
L\K. O
Remark. Note that the above argument does not work for geometric kequivalence (see
end of 2) in place of kquasiisotopy.
Since Ck = ak (mod gcd(co,..., Ck1)), Theorem 3.4 implies the case 1 1, and also
the 2component case of
Theorem 3.5. [26] Set A = 1) 1)]. The residue class of CA+k modulo gcd of
CA, ..., CA+k1 and all pinvariants of length < I is invariant under (L[ + k)quasi
isotetl'r
(Here [x] = nifx E [n n + ), and Lx] = n ifx E (n ,n+ ] for n Z.)
One special case not covered by Theorem 3.4 asserts that for 3component links the
residue class of Ck modulo the greatest common divisor Ak of all pinvariants of length
< k + 1 is invariant under [L quasiisotopy. Naturally, one could wonder whether an
integer invariant of [L quasiisotopy of 3component links, congruent to Ck (mod Ak), can
2~YCD D~~ IV~IIVII~II~)~I~UI~~ I
be found among coefficients of rational functions in V* of the link and its twocomponent
sublinks. It turns out that this is not the casealready for k = 1. Indeed, we would get an
integer link homotopy invariant of (monochromatic) finite type (specifically, of type 4),
which is not a function of the pairwise linking numbers (since cl p(123)2 (mod Ai), cf.
[18, 23]). But this is impossible for 3component links [24].
Alternatively, one can argue directly as follows. Consider the invariant 7(L) :
at(L) E ao(Lo)ai(Li), where the summation is over all ordered pairs (Lo, L) of
distinct 2component sublinks of L. (Recall that in the 2component case ao(L) and al(L)
coincide with the linking number and the generalized SatoLevine invariant, respectively.)
It can be easily verified that 7(L) jumps by
p(12) ((1, 3+)(2, 3) + (1, 3(2, )+(, )(23+))
on any singular link L, K1 U K2 U K, with smoothing K1 U K U K3+ U K3 One
can check that this jump cannot be cancelled by the jump of any polynomial expression in
co(L) and the coefficients of the Conway polynomials of the sublinks of L, homogeneous of
degree 4.
Remark. An integer invariant of link homotopy of 3component links is given by qA1 + r,
where q is any polynomial in 1JI) and r p (123)2 (mod A1), 0 < r < A1.
Al
CHAPTER 4
THE MULTIVARIABLE ALEXANDER POLYNOMIAL
The Conway polynomial is equivalent to the monochromatic case of the Co i' ,.';
potential function QL of the colored link L, namely VL(x x1) = (x X 1)QL() for
monochromatic L. For a link L colored with n colors, 2L E Z[x"1,. ,1, X] (the ring of
Laurent polynomials) if L has more than one component; otherwise QL belongs to the
fractional ideal (x x1)1 [x"1] in the field of fractions of Z[x1].
fQL(XI,... X,) is a normalized version of the signrefined Alexander polynomial
AL(tl,..., tn), which is welldefined up to multiplication by monomials tl ... t^. If K
is a knot, (x x1)QK(x) = AK(x2), whereas for a link L with m > 1 components,
QL(x1,...,Xn) = X ... "AL(x(,.L..,Xc(n)), where c: {1,...,m} {1,...,n} is the
coloring, and the integers A, A1,..., AT are uniquely determined by the symmetry relation
QL(x1,... ,Xn) = (1)mL(xi ,... ,x~'). We refer to Hartley [11], Traldi [35] and Turaev
[36] for definition of the signrefined Alexander polynomial and discussion of the Conway
potential function.
Remark (not used in the sequel). The .,vmmetry of the Alexander polynomial, which
forces one to work with its symmetrized version fL, goes back to the .ivii. l. I ry of
presentations of the Alexander module or, equivalently, of the Fox differential calculus.
The Leibniz rule for the Fox derivative (restricted to the group elements)
D(fg) = D(f) + fD(g)
arises geometrically from considering lifts of the generators of the fundamental group
of the link complement to the universal abelian cover, starting at a fixed lift p of the
basepoint. If we wish to base the whole theory on the lifts ending at p, we will have to
change the Leibniz rule to
D(fg)= D(f)g + D(g).
The "symmetric" Leibniz rule
D(fg) = D(f)g1 + fD(g) + (f'g1 gf1)
does not seem to have a clear geometric meaning. (Note the analogy with the three first
order finite differences.) However, it does correspond to a "symmetric" version of the
Magnus expansion, which will be behind the scenes in this section. Indeed, its abelian
version gives rise, in Theorem 4.2 below, to a power series SL, in the same way as the
abelian version of the usual Magnus expansion leads to Traldi's parametrization of QL,
discussed in the proof of Theorem 4.8 below.
Lemma 4.1. (compare [15]) (a) L(x1, X) QL(X1 ,... , 1).
(b) If L has m > 1 components, the total degree of every nonzero term of QL is
congruent to m (mod 2).
(c) If L has > 1 components, the xidegree of every nonzero term of QL is congruent
mod2 to the number ki of components of color i plus li := lk(K, K'), where K runs over
all components of color i and K' over those of other colors.
Proof. It is wellknown that, if all components have distinct colors, the integer Ai from the
above formula relating QL to the Alexander polynomial is not congruent to li mod 2 (cf.
[35]). This proves (c), which implies (b). For links with > 1 components (a) follows from
(b) and from the symmetry relation in the definition of QL; for knots from the relation of
QL(Z) with the Conway polynomial. O
Computation of QL is much harder than that of VL, for the skein relation (1) is no
longer valid for arbitrary crossing changes. It survives, in the form
L+ L (Xi 71))Lo, (2)
in the case where both strands involved in the intersection are of color i. There are other
formulas for potential functions of links related by local moves, which suffice for evaluation
of Q on all links; see references in Murakami [29].
Dynnikov [6] and H. Murakami [29] noticed that the coefficients of the power series
QL(eh /2, .. ., C /2) are (monochromatic) finite type invariants of L.
Theorem 4.2. For a link L colored with n colors there exists a unique power series
1SL E Q[[zi,... z]] (unless L is a knot, in which case 3L E z'Z[z]) such that
OL(X1 X1 1,... Xn X ) = L(x, ... ,Xn).
The coefficient of 1L at a term of total degree n is of ';. n + 1; moreover, 13L './.:'/f the
skein relation
3L 1L L= ZiLo
for intersections between components of color i. Furthermore, the total degree of ev
ery nonzero term of ZL is congruent mod2 to the number of components of L, and
UL(4yi,... ,/ ) E Z[[yi,... y]] (unless L is a knot).
For n = 1 we see that ZSL(z) coincides with the (finite) polynomial VL(z).
Proof. Let x(z) denote the power series in z, obtained by expanding the radical in either
of x(z) = + 1 + z2/4 + z/2 by the formula (1 + t)r 1 + rt + rt2 + .... The Galois
group of the quadratic equation z = x x acts on its roots by x x1, so Lemma
4.1(a) implies that the coefficients of the power series
UL(ZI,... ,Z.) := L(X(Zl),...X ,X(Z))
are independent of the choice of the root x.
The second assertion follows by Murakami's proof [29] of the result mentioned above,
and the third from the skein relation (2). Next, since x1 x = (x x),
UL(ZI,...,z,)= L(X(Zl)1, ..,x(z,)1)
= ()m l(x(l), ,x(,))= ()m'l (I, ... n,),
which proves the assertion on total degrees of nonzero terms.
The final assertion may be not obvious (since (k + 1) { (2 1) for k = 1,3, 7) from
(t 4+( )k (2k t)y.
(1+4y)1/2 +4Z k + t k 1Nk
k 0
However, this power series does have integer coefficients, since this is the case for
2k 1
(l+4y)1/2 + 2 )k(2 k k )k.
k= /
(In fact, we see that (k + 1)  2(2k1), and x(y) 1 (mod 2).) D
Let us consider the case of two colors in more detail. Let ci denote the coefficient
of 3L(zl, Z2) at ziz2. Since 63(z, z) = V(z), coo coincides with the linking number co
in the 2component case and is zero otherwise. The skein relation (2) implies that for a
3component link K1 U K' U K2, colored as indicated by subscript, c0o coincides with
lk(Ki, KI)(lk(Ki, K2) + lk(K,, K2)) up to a type 0 invariant of the colored link. The latter
is identically zero by the connected sum1 formula
OL#iL' (xj X1 (3)
which follows from the definition of OL in Har'!. [11]. Since col = co clo or by an
analogous argument, col = lk(K1, K2) lk(K/, K2). It follows that for 2component links
cl coincides with the generalized SatoLevine invariant a, up to a KLtype 0 invariant.
Since a, cl is of type 3, it has to be a degree 3 polynomial in the linking number, which
turns out to be T(lk3 k). (Since Ounlink 0 and fHopf link 1, the second Conway
identity [11, 15] can be used to evaluate OL on a series of links 'H, with IkH, n.) The
halfinteger cl is thus the unoriented generalized SatoLevine invariant, which is implicit
in the second paragraph of Kirk and Livingston [16]. Specifically, 2c1n/lk2 is (unless
1k = 0) the CassonWalker invariant of the Qhomology sphere obtained by 0surgery
on the components of the link; in fact, cn(L) = a()+l(L') where L' denotes the link
obtained from L by reversing the orientation of one of the components [16].
The preceding paragraph implies the first part of the following
1 The multivalued operation of Hashizume connected sum L#jL' is defined as follows.
Let L'(mS1) be split from L(mS1) by an embedded S2, and b: Ix I l S3 be a band meet
ing the embedded S2 in an arc, and L(mS1) (resp. L'(mS1)) in an arc of color i, identified
with b(I x {0}) (resp. b(I x {1})). Then L#iL'(mS') ((L(rmS) U L'(mS')) \ b(I x 01)) U
b(0I x I).
Theorem 4.3. Consider the power series
/, SL(zl, ..., z,)
V (zc(1)) V (zc())
where K1,..., Km denote the components of L. Let cij denote the coefficient of 3*L(zi, z2)
at z'z'. For 2component links
(i) aoo = lk, and all is the unoriented generalized SatoLevine invariant, which
assumes all halfinteger values;
(ii) Oi,2k1 and (. 1,1 coincide, when multiplied by (1) k+1, with Cochran's [5/
derived invariants P3, whenever the latter are 1. fI,' (i.e., 1k = 0);
(iii) aij is of type i + j; when i + j is odd, ij = 0.
The last part is an immediate corollary of Theorem 4.2.
Proof of (ii). It follows from [5] that C i / (z2i iL= L(2), where rTL is Kojima's
yfunction and z = y y1. On the other hand, it follows from [13] that
(1 y2)(1 y)Q (1, y)
VK2( y 1)
where Q' is given by QL(X, X2) = (X1 x'1) 2 1) (Xl, X2) for 2component links
with 1k = 0. It is shown in Jin [13] (see also proof of Theorem 4.8(ii) below) that under
this assumption Q' is a Laurent polynomial (rather than just a rational function). Define
3' by UL(zi, z2) z IZ2 LZI, Z2) for 2component links with 1k = 0, then
S z 2'/(0, z)
(1) z Vi() '
i 7K (z)
and consequently
( 1)fi+21z2i L(O, z)
(O K,1 (0)VK, (z)
According to (i), the sign must be positive. E
It turns out that each coefficient of the power series 1 can be canonically split into a
Qlinear combination of the coefficients of certain 2"1 polynomials.
23
Theorem 4.4. The Com, ri;, potential function of ,;,1 colored link L with > 1 components
can be ;,,:.,;,. l;i written in the form
QL(x1,..., Xn) {= "i } i1...i2k ({xi},..., {x }) (4)
l il<...
for some Pil...i2k e {Z[ 1,' .I.., ,z], where {f(xl,... x)} denotes f(xl,..., x,) +
f(x1 .,..., x1 ) for ,;; function f(xl,... ,x,). Moreover, if n = 21, the coefficients of
P12...n are integer.
Here is the case n = 2 in more detail:
Q(x, y) 2P(x x, y y) + (xy1 + xy)P12(x x,y y).
This case was essentially known in 1986; indeed it is equivalent (see formula (*) below) to
Kidwell's decomposition [15]:
Q(x, y) = Ki(x x y y1) + (xy + x y1)K2(x X_, y y1).
However, the proof of Theorem 4.8(iii) below (on pinvariants) breaks down for K1 and K2
in place of 2P and P12. Moreover, the assertion on integrality of 2Pi, 2 in Theorem 4.4
will not hold already for n = 3 (resp. n = 5) if {I 1} is replaced with {xl, x2,
Xz2
(resp. with { }) in (4), at least for some Laurent polynomial Q satisfying the
+1
conclusion of Lemma 4.1(a). (Lemma 4.1(a) is the only property of fL used in the proof
of Theorem 4.4.)
Proof. By Lemma 4.1(a), fL includes together with every term Ax"' .. xTP the term
(1)P1++PAxiPI ... xnn and so can be written as a Zlinear combination of the L
polynomials {xl ... x }. The formula
{xiM} {x 1M} = {x}{M}, (**)
where M is any monomial in x 1,..., can be verified directly (separately for M of
odd and even total degrees), and allows to express each {x1 ... xn } in the form
1 i { X 1 ( x ) 1) i}, {x,})k
lii<...
for some P.i EZ Z[i, ., z], k > 0, and some P' C [E [:;, .. z]. The summands
corresponding to k = 1 can be included in P', and one can get rid of the summands
corresponding to odd k > 3 by repeated use of the formula
2{xx 'M}= {x}{xf M} + {x '}{xM} + {xjxj}{M},
combined with (*). Note that (**) follows immediately from (*) and its analogue
{xiXjM} + {x 'x M} = {xXj}{M}.
But it is not clear from this approach that the resulting polynomials have halfinteger
coefficients. To see this, represent 2{xijx 2Xi3 ... x2 i2k+} as
{ I r  I X7 I I ... '71 1Y' \
{xlx2x i3 Xi2+l } 7 l 22k+l IXJ
2k
 r1 1 .)3 1 i2k
+j 1 1 i ij +l Xj+3 72 Xfe+l
j1
Then formula (*) yields:
I21Xi3 X XI2 I
2k+1
1 j+ i ir i  1 r 1 i 1 i 1
( Jl{x }{x lx x ... (x 2Xij IX'+lXij+2X +3) ?211) 2k ~k .
j=1
Thus each P .i' 2 can be included in the polynomials P .
Thus each 21.2
It remains to verify uniqueness of (4). Suppose, by way of contradiction, that a
nontrivial expression Q(xi,..., x,) in the form of the right hand side of (4) is identically
zero. Then so is Q(xl,..., Xn,_, x) Q(xl,... X 1, x ), which can be rewritten as
[xn] [x x x7~I 3 1 i2k1 2k 1 Pil*...i2k1,n({xi},..., {xn}) 0,
1,il<...
where [f(xi,... x)] denotes f(xl,... x) f(x 1,..., x). Denote the left
hand side by [x,]R(xl,..., x,), then R(xl,...., x,) is identically zero. Hence so is
R(xl,..., x,_2, x1, xn) R(1,... x_ n2, nl, x), which can be rewritten as
[x_l]
1 ," '
S {XiX Xi2k lXi }Pil...i2k,n,n({Xl},..., {Xn}) = 0.
1 il<
(**)
Repeating this twostep procedure ['] times, we will end up with
[xi]{1}Pi...n({xi},... {xn}= 0 or [x21{}P2...n({xi}, ,{x ) = 0
according as n is even or odd. Consider, for example, the case of odd n. By symmetry,
P1..... = 0 for each i. Returning to the previous stage
[X4] i i 1 i 1 } il...i2...k,4..n({1. {Xn}) = 0,
1il1<
we can now substitute zeroes for P23,4...n, P13,4...n, P12,4...n, and so we get P4...n 0.
Continuing to the earlier stages, we will similarly verify that each P,...i = 0. E
Since every term of, , x x1 has an odd degree in x, Lemma 4.1 implies
Lemma 4.5. For every nonzero term of Pil...i,
(i) the total degree is congruent mod2 to the number of components;
(ii) the zidegree is congruent to ki + li (mod 2) iff i {i,..., i,}.
Let us turn again to the case of twocomponent links colored with two colors. Let d'
(resp. d"j) denote the coefficient of P(z, z2) (resp. P12 Z2)) at z'z. Substituting x(zi)
for xi as in the proof of Theorem 4.2, we get
2 2
1 ziz2 I Z2
rx2 1 + x1 2 2 + + + terms of total degree > 4.
2 4 4
Thus coo = 2(doo + d"o) and
c = 2(d', + d,) C2 2(do + d2l) + 2 = 2(dO2 + d2) +
2 4 4
By Lemma 4.5(ii), d2i2j d+1,2j+1 = 0 if the linking number is even; otherwise
di+1,2j+1 di2j = 0. On the other hand, coo = k and cn = a (1k3 lk)/12 by Theorem
4.3(i). Thus 2d"g is 1k or 0, and 2d", is 0 or the integer ac (1k3 lk)/12, according as 1k
is even or odd. Similarly, 2doo is 0 or 1k, and 2d' is the integer ac (1k3 41k)/12 or 0,
according as 1k is even or odd. On the other hand, it follows from Proposition 2.1 that the
KLtype 0 invariant 2doo = ((1)lk + 1) k /2 is not of finite type. Since the linking number
coo and the halfinteger cl are of finite type, the integer d1 := 2(d' + d',) is also not of
finite type.
Remark. The decomposition (4) in the case n = 2 can be modified, using the identity
S (x y)2
xy1 + x y = y + 2,
xy
to the form
L(, y) 2(P({x}, {y}) + P12({ {y})) + (X P12( {y}).
xy
Lemma 4.5(ii) implies that each coefficient of P + P12 coincides with the corresponding
coefficient of either P or P12, depending on the parity of lk(L). Since lk(L) is the constant
term of 2(P + P12), the remainder (xy)l(x y)2P12 is redundant, in the sense that it
contains no additional information with respect to 2(P + P12).
More generally, we have
Theorem 4.6. Let Pi, ..., i2 be the p.'l'. I,;;,',.*l of the colored link L, /. I,.. l in (4). The
" ,;;,,. .;;,,d 7V L, f .u, ,l by
VL(zI,...,z ) := 2 Pi ..., (z ,..., ) e Z[zi,..., zn]
l
if L has at least two components, and by VL(z) = z1VL(z) E z'Z[z] if n 1, determines,
and is determined by, the Co ,i, ia ; potential function QL (X, ,xT,). The coefficient of VL
at a term of total degree d is of colored type d + 1, and vanishes unless d is congruent mod2
to the number of components of L.
Proof. By Theorem 4.4, 2L determines each Pi~..., hence VL. Conversely, Lemma 4.5(ii)
implies that the coefficients of each Pi~...,i2k hence of QL, are determined by VL. The last
two assertions follow from Lemma 4.5(i) and the skein relation (2), which can be rewritten
in the form
VL+ VL = ZiVLo
for intersections between components of color i.
Let K1,... Km denote the components of a colored link L, where m > 1. It follows
from Theorem 4.4 that the power series
0,*(n r_L(z I,. ,an) _. 1 il
Q (Xi,... ,Xn) CK 1 (7)[[ L(,.X.. ,a) 1]
VKi (Xc(i) Xc(1)) VKr (Xc(mn) c(m)
can be expressed in the form
(x ,... ,x ) { }il ({xl},.. x }),
Sil<..
where each P., E Z[[zi,..., z,]], and the proof of Theorem 4.4 shows that such an
expression is unique. The doubled sum of all these polynomials is, of course, nothing but
*(z1, i,... z,)
V2(Z1,... ,Zn) :=
S : VK, (Zc(1)) VKm (Zc(mn))
which contains the same information as QT by the proof of Theorem 4.6.
Corollary 4.7. (i) For each L: mS1 S3 and ,;1 k E N there exists an Ek > 0 such
that if L': mS1 S3 is CO Ekclose to L,
V, = V + terms of total degree > k,
3*, = 1* + terms of total degree > k.
(ii) V and 1* can be ;i,', '.:. 1; extended to all TOP links in S3, preserving (i).
(iii) The extended VL and S5* are invariant under TOP isotei,; of L.
Proof. As in the above argument for V1, it is easy to see that VL, as well as S, are
invariant under PL isotopy (cf. [35]). By Theorems 4.2 and 4.6, the coefficients of VL and
13 at terms of total degree k are of colored type k + 1, hence by Theorem 2.2 they are
invariant under (k + 1)quasiisotopy. The rest of the proof is as in Theorem 3.2. D
Remark. If the components of L are unknotted, or more generally have no nonlocal knots,
VL is a (finite) polynomial. Actually, VL splits into a product of m onevariable power
series and a polynomial (namely, the product of the irreducible factors of VL, involving
more than one variable), so that each of them is individually invariant under PL isotopy;
compare [35, 32]. There is, of course, such a splitting for every Pi...i producing a
plethora of PL isotopy invariants.
Lack of an analogue of Lemma 4.5(ii) for the power series 1L hinders establishing a
simple relation between the coefficients of 1L and those of Traldi's expansion of QL [35],
which prevents one from expressing the finite type rationals yij (discussed in Theorem 4.3)
as lifting of pinvariants. However, each cij can be split into a linear combination (whose
coefficients depend on the linking number) of the integer coefficients of the power series
VL, which do admit such an expression.
Theorem 4.8. Let Sij denote the coefficient of VL at zIz For 2component links
(i) 6oo = Ik, and 6 is an integer 1':f1.:',, of the SatoLevine invariant p(1122), but not
a finite ';I'p. invariant;
(ii) 61,2k1 and 62k1,1 are Cochran's [5/ derived invariants (1) +10 whenever the
latter are / I/., l (i.e., 1k = 0);
(iii) 6ij is an integer '.:fl.:.', of Milnor's i.'] invariant (1)j (1. .12... _2), provided
i+1 j+1
that i + j is even;
(iv) 6ij is of KL',,I,. i + j;
(v) when i + j is odd, ij = 0;
(vi) when i + 1, j + 1 and 1k are all even or all odd, 6ij is even;
(vii) for a given L, there are only finitely i,,,.'i; pairs (i,j) such that yij 0 0 modulo the
greatest common divisor Aij of all kl 's with k < i, 1 < j and k + 1 < i + j; the congruence
can be replaced by i./;,.';:1;/ if the components of L are unknotted.
The last four parts are immediate consequences of Theorem 4.6; Lemma 4.5(i);
Lemma 4.5(ii) and integrality of the coefficients of P,*2; the definition of VL as a rational
power series. The first part follows from the discussion after Lemma 4.5.
Remark. The geometry of the relationship between pinvariants and the multivariable
Alexander polynomial is now better understood [17]. If L is the link closure of a string
link f: {1,..., m} x I R2 x I,
AL AjF,
where AL denotes the usual Alexander polynomial, A5 the Alexander polynomial of
the string link [19], i.e., the Reidemeister torsion of the based chain complex of the pair
(X, X n R2 x {0}), where X = R2 x I \ im and Fe a certain rational power series,
determined by the pinvariants of f.
Proof of (iii). Since we are only interested in the residue class of every 6ij mod Aij, we
may consider the polynomial VL 2(P+P12) in place of the power series L = 2(P*+P*2).
By Theorem 4.4 and Lemma 4.5(ii), 2L is uniquely expressible in the form
({Xl}{X2})2 L(Xl, X2) Ql1({}2, x2}2) + {X121}{XI}{1X2}Q({XI}2, {2}2)
for some Q1,Q2 C Z[i,Z21, where A = 0 or 1 according as lk(L) is odd or even. Set
y, = x2 1, then x2 = 1 Yi + y. y + .... The identities
{xjXx}{xIX} {2 (X2 + 2)(x )(X 1) (2 yIy2 + Y+Y )Y12
allow to express (x1x2)AL(X1, x2) as a power series TL(yI, Y2) with integer coefficients.
Let us study this substitution more carefully. Let d.j, d' denote the coefficients
at z'z' in P and P12, and let us write (k,l) < (i,j) if k < i and / < j. Then the
coefficient ej at yy in the power series R2, defined by the equality (yIy2) 'R2(yi, Y2)
(ziz2)A1Pl2(zl, z2), is given by
e4 (1)(ik)+ (j1)d (_)'+j k
(k,l) (ij) (k,l) (ij)
(the latter equality uses that d"' = 0 if k + 1 is odd), and similarly for the coefficients ei of
the power series R1, defined by (yiy2)Ri(yi, Y2) = (Z1i2)P(Z1I z2). Now the coefficients
eij of TL(y, y2) are given by
ey 2e + 2e%' + ( )k + (1)j1 t
k
2e + 2e' + ( )+ > ((i k) + (j )>/',
(k,1)<(ij)
The key observation here is that all coefficients on the right hand side are even if i + j is
even.
Let us consider the case A = 1. Then by Lemma 4.5(ii), d' = 0 unless both i and j
are even, and dj 0 unless both i and j are odd. Hence e% = 0 modulo gcd{e" (k, 1) <
(i,j)}, unless both i and j are even, and similarly for e Now it follows by induction that
eij 2e' 2d' or eij 2e.j = 2dj modulo Eij : gcd{ek  (k,1) < (i,j)} according
as i and j are both even or both odd. Thus ej 2(dj + d%) (mod Ejy) if i + j is even.
Clearly, the latter assertion holds in the case A = 0 as well, which can be proved by the
same argument.
Finally, since A Ik k1 (mod 2), and xn 2 are expressible as power series in yi
with integer coefficients and constant term 1, the coefficients eij of Traldi's power series
TL(yl, y2) (X2)lkl IL are related to eij by congruence modEyi. This completes the
proof, since by Traldi [35], each eij is an integer lifting of (l) +1'(1...12... 2).
i+1 j+i
Remark. The above argument yields a new proof of another result due to Traldi:
2p(1...12... 2) 0 when i + j is odd [34].
i+1 j+i
Proof of (ii). We recall that p(1... 12) and p(12... 2) identically vanish, with the excep
tion of p(12) Ik [28]. Hence by (iii), o0,2k 62k,0 0 (mod 1k) for each k. So if Ik = 0,
every nonzero term of either P* or P,2 involves both zl and z2. (Alternatively, this follows
from Jin's lemma mentioned in the proof of Theorem 4.3(ii).) By Lemma 4.5(ii), every
nonzero term of P2 has to further include each of them once again, i.e., P*2 is divisible
by z1z2. Hence, firstly, 62k1,1 coincides with the coefficient of 2P*(zi, z2) at z k 2, and,
secondly, this coefficient is not affected by adding (x(zi)x(z2)1 + x(z2)x(x) 1)P2(i, z2)
to 2P*(zl, z2), where x(z) is as in the proof of Theorem 4.2. O
Theorem 4.9. The coefficient of V at z ... z is invariant under kquasiisote':,, if
max(i,..., in) < 2k.
Proof. This is analogous to Theorem 3.4. One only needs to show that the coefficient
in question is invariant under restricted skein kquasiisotopy, where a restricted skein
kquasiembedding is defined as before, but with additional restriction that each Loi2...,ik
includes L \ K and each Ho2... i is fixed on L \ K. This is done by the same argument,
replacing every occurrence of the polynomial zImVL with VL. E
Corollary 4.10. (a) [26] Cochran's invariants /3 are invariant under kquasiisot(i',,
(b) Milnor's invariants p(1 ... 12... 2) of even length are invariant under kquasi
isot(',,i if each index occurs at most 2k + 1 times.
Part (b) covers (and largely improves) the corresponding case of
Theorem 4.11. [261 All pinvariants of length < 2k + 3 are invariant under kquasi
isot ,i,,i
CHAPTER 5
THE HOMFLY AND KAUFFMAN POLYNOMIALS
We recall that the HOMFLY(PT) P y. 1,;;,,..',,:l and the Dubrovnik version of the
Kauffman F ''.;; ,,ii.:,,l1 are the unique Laurent polynomials HL, FL cE Z[x1, y1] satisfying
Hunknot = unknot = 1 and
XCHL x1HL = yHLO,
XFL  XFL Y(FLo xWw(L)u(FLo),
where L+, L_, Lo and L8 are as in the definition of the Conway polynomial (cf. 3), L, is
obtained by changing the orientation of the iight" of the two loops in L8 (corresponding
to either the two intersecting components or the two lobes of the singular component)
and oriented smoothing of the crossing of the obtained singular link L', and w(L) denotes
the writhe of the diagram of L, i.e., the number of positive crossings minus the number of
negative crossings (so that w(L+) 1 = w(L_) + 1 = w(Lo)). The versions of HL and FL
in Lickorish [20] are obtained as HL(ia, iz) and (1)"1FL(ia, iz).
Theorem 5.1. Let et denote the (formal) power series 7 and consider the power
series
HL FL
HL := and FL :=
HK1 ... HK and FK1 ... FK,
where K, Km denote the components of the link L.
(i) For each L: mS1 c S3 and 1,;1 n E N there exists an F,, > 0 such that if
L': mS1 c_ S3 is CO Eclose to L,
H,(eh/2, e/2 h/2) H(ech/2,eh/2 h/2) mod (he),
FL,((c1)h2, eh/2 h/2) F (C(c1)h/2, h/2 h/2) mod (h).
(ii) HL and FL can be ;,,::.,;,. 1: extended to all TOP links in S3, preserving (i).
(iii) The extended H1 and FL are invariant under TOP isote'i' of L.
Proof. The connected sum formulae for HL and FL [20] imply that H* and F* are
invariant under PL isotopy. (Note that the connected sum in Lickorish [20] is Hashizume's,
not the componentwise connected sum of Melikhov and Repovs [25].) On the other hand,
it was noticed in Lieberum [21] (compare [9, 4, 33]) that the coefficients of the power series
oo k+m1
HL(ech/2, C h/2 h/2) k pi k 1
ll( d / h) Pki c' c Q[c]l[[hl]]
k=0 i=0
oo k+m1
FL(e(c1)h/2, Ch/2 2h/2) iC Ck 1
> qkich (Q[c][[h]]
k=0 i=0
are (monochromatic) finite type invariants of L. Specifically, each pki and each qki is of
type k, moreover oi = qo = 6,m,i (the Kronecker delta). (The argument in Lieberum [21]
was for HLHT2 and FLFT2, where T2 denotes the trivial 2component link, but it works as
well for HL and FL, compare [3].) The rest of the proof repeats that of Theorem 3.2. O
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of Manifolds and Varieties, Adv. Soviet Math. 18, Amer. Math. Soc., Providence RI
(1994), 167172.
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11 R. Har',v, The Conway potential function for links, Comm. Math. Helv. 58 (1983),
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121 J. R. Hughes, Finite / 1,'. link homot(./,' invariants of ktrivial links, J. Knot Theory
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Math. Soc. 284 (1984), 401424.
[35] L. Traldi, Conway's potential function and its Taylor series, Kobe J. Math. 5 (1988),
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V. Turaev, Introduction to combinatorial torsions, Birkhauser, Basel (2001).
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BIOGRAPHICAL SKETCH
I was born in Moscow, Russia, in 1980.
My high school was the Moscow State FiftySeventh School, where I studied in a
mathematically oriented class in 199296.
In 19962001 I continued my education at the Mechanics and Mathematics Depart
ment of the Moscow State University, which conferred a bachelor's degree in pure and
applied mathematics on me.
I have been in the Graduate School of the University of Florida since 2001.

PAGE 2
IamindebtedtoProf.A.N.Dranishnikovforhisencouragementandguidancethroughmystudyinthegraduateschool,toProf.D.Cenzer,Prof.J.Keesling,Prof.S.ObukhovandProf.Yu.B.Rudyakfortheirinterestinmydissertation,andtothefacultyoftheMathematicsDepartmentfortheirhospitality.Last,butnotleast,IwouldliketothankDr.P.M.AkhmetievforstimulatingcorrespondenceanddiscussionsandJuanLiuforhersupportandpatience. ii
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page ACKNOWLEDGMENTS ................................. ii ABSTRACT ........................................ iv CHAPTER 1INTRODUCTION .................................. 1 2COLOREDFINITETYPEINVARIANTSANDkQUASIISOTOPY ...... 3 3THECONWAYPOLYNOMIAL .......................... 10 4THEMULTIVARIABLEALEXANDERPOLYNOMIAL ............ 18 5THEHOMFLYANDKAUFFMANPOLYNOMIALS ............... 32 REFERENCES ....................................... 34 BIOGRAPHICALSKETCH ................................ 37 iii
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iv
PAGE 5
v
PAGE 6
RecallthatPLisotopyoflinksinS3canbeviewedastheequivalencerelationgeneratedbyambientisotopyandinsertionoflocalknots.(ThusanyknotinS3isPLisotopictotheunknot.)The\linksmoduloknots"versionoftheVassilievConjectureforknots[ 37 ]couldbeasfollows. 25 ]andTheorem2.2below),whereastheoriginalVassilievConjecturefailsforknotsinanysuchWembeddableinR3(e.g.,intheoriginalWhiteheadmanifold)[ 7 8 ].Similarly,ifMisahomotopysphere,thenM=S3giveneitherapositivesolutionto1.1forknotsinM([ 25 ]andTheorem2.2below),oraproofoftheVassilievConjectureforknotsinM#M[ 7 8 ]. SomefurthergeometrybehindProblem1.1isrevealedby (i)nitetypeinvariants,welldeneduptoPLisotopy,arenotweakerthanniteKLtypeinvariants(cf.x2),welldeneduptoPLisotopy; (ii)indistinguishabilitybyniteKLtypeinvariants,welldeneduptoPLisotopy,implieskquasiisotopyforallk2N; (iii)kquasiisotopyforallk2NimpliesTOPisotopyinthesenseofMilnor(i.e.,homotopywithintheclassoftopologicalembeddings); (iv)TOPisotopyimpliesPLisotopy.
PAGE 7
Theassertionfollowssincetheconversetoeachof(i){(iv)holdswithnoproviso:to(ii)byTheorem2.2below,to(iii)byTheorem1.3(b)below,to(i)and(iv)bydenitions. WerecallthatPLlinksL;L0:mS1,!S3(wheremS1=S11ttS1m)arecalledkquasiisotopic[ 25 ],iftheyarejoinedbyagenericPLhomotopy,allwhosesingularlinksarekquasiembeddings.APLmapf:mS1!S3withpreciselyonedoublepointf(p)=f(q)iscalledakquasiembedding,k2N,if,inadditiontothesingletonP0:=ff(p)g,thereexistcompactsubpolyhedraP1;:::;PkS3andarcsJ0;:::;JkmS1suchthatf1(Pj)Jjforeachj6kandPj[f(Jj)Pj+1foreachj
PAGE 8
Weintroducenitetypeinvariantsofcoloredlinks,asastraightforwardgeneralizationoftheKirk{Livingstonsettingofnitetypeinvariantsoflinks[ 16 ].Kirk{Livingstontypekinvariantsarerecoveredastypekinvariantsofcoloredlinkswhosecomponentshavepairwisedistinctcolors,whereastheusualtypekinvariantsoflinks(asdenedbyGoussarov[ 9 ]andStanford[ 33 ])coincidewithtypekinvariantsofmonochromaticlinks.Althoughnormallyeverythingboilsdowntooneofthesetwoextremecases,thegeneralsetting,apartfromitsunifyingappeal,maybeusefulinproofs(cf.proofofProposition3.1)andindealingwithcoloredlinkpolynomials(cf.Theorem4.6).Refertothefollowingbracketedcitationsforabackgroundonnitetypeinvariants:[ 37 10 31 4 33 3 ]. Anmcomponentclassicalcoloredlinkmapcorrespondingtoagivencoloringc:f1;:::;mg!f1;:::;mgisacontinuousmapL:mS1!S3suchthatL(S1i)\L(S1j)=;wheneverc(i)6=c(j).Thususuallinkmapscorrespondtotheidentitycoloringandarbitrarymapstotheconstantcoloring.LetLM31;:::;1(c)bethespaceofallPLclinkmapsmS1!S3,andletLM(c)denoteitssubspaceconsistingofmapswhoseonlysingularitiesaretransversaldoublepoints(theintegermwillbexedthroughoutthissection).NotethatLM(c)isdisconnectedunlesscisconstant.LetLMn(c)(resp.LM>n(c))denotethesubspaceofLM(c)consistingofmapswithprecisely(resp.atleast)nsingularities.NotethatLM0(c)doesnotdependonc. Givenanyambientisotopyinvariant:LM0(c)!GwithvaluesinanitelygeneratedabeliangroupG,itcanbeextendedtoLM(c)inductivelybytheformula(Ls)=(L+)(L); 3
PAGE 9
withatleastr+1transversaldoublepoints,forsomeniter,theniscalledanitectypeinvariant,namelyofctyper.Notethatinournotationanyctyperinvariantisalsoofctyper+1.Sometimeswewillidentifyanitectypeinvariant:LM0(c)!Gwithitsextension:LM(c)!G. Form=1thereisonlyonecoloringc,andnitectypeinvariants,normalizedby(unknot)=0,coincidewiththeusualGoussarov{Vassilievinvariantsofknots.Ifcisaconstantmap,wewrite\type"insteadof\ctype",andifcistheidentity,wewrite\KLtype"insteadof\ctype",andLMinsteadofLM(c).Ifthecoloringcisacompositionc=de,thenanydtypekinvariantisalsoactypekinvariant.Inparticular,anytypekinvariantisaKLtypekinvariant,butnotviceversa.Indeed,thelinkingnumberlk(L)andthegeneralizedSatoLevineinvariant(seex3)areofKLtypes0and1,respectively,butoftypesexactly1and3. ItisarguablethattripleandhigherinvariantswithdistinctindicesshouldberegardedashavingKLtype0(astheyassumevaluesindierentcyclicgroupsdependingonlinkhomotopyclassesofpropersublinks,wemusteitheragreetoimaginethesegroupsassubgroups,say,ofZQ=Z,orextendthedenitionofniteKLtypeinvariantstoincludethepossibilitywhereeachpathconnectedcomponentofLMmapstoitsownrange).Incontrast,therearealmostnonitetypeinvariantsoflinkhomotopy[ 24 ],althoughthiscanberemediedbyconsiderationofstringlinks(seereferencesinHughes[ 12 ]orMasbaumandVaintrob[ 23 ])orpartiallydenednitetypeinvariants[ 9 12 ].Weshowinx4thatcertaininvariantsof2componentlinkshaveintegerliftingsofniteKLtype,atleastsomeofwhicharenotofnitetypedueto Proof. ItisclearthatKLtyperinvariants:LM0!Gformanabeliangroup,whichwedenotebyGr(orG(m)rwhenthenumberofcomponentsneedstobespecied).Clearly,G0isthedirectsumofj0(LM)jcopiesofG.LetGr()denotethesubgroupofGrconsistingofinvariantsvanishingonalllinkswhoselinkhomotopyclassisnot20(LM),and
PAGE 10
onaxedlinkB2withunknottedcomponents.ThelattercanalwaysbeachievedbyaddingaKLtype0invariant,sothequotientmapq:Gr!Gr=G0takesGr()isomorphicallyontoasubgroup,independentofthechoiceofB;moreoverGr=G0=Lq(Gr()).Foranylk2Z,itwasprovedinKirkandLivingston[ 16 ]thatZ(2)1(lk)'Z(generatedbythegeneralizedSato{Levineinvariant)andconjecturedthatZ(2)r(lk)isnotnitelygeneratedforr>1.Itiswellknown[ 33 ]thatthegroupofmonochromaticGvaluedtyperinvariantsofmcomponentlinksisnitelygenerated;inparticular,soisG(1)r.Let~GrdenotethesubgroupofGrconsistingofthoseinvariantswhichremainunchangedundertyinglocalknots,i.e.,underPLisotopy.Noticethatthewellknownproofthattype1invariantsofknotsaretrivialworksalsotoshowthat~G1=G1.InthefollowingremarksweaddressthedierencebetweenGrand~Grforr>1.
PAGE 11
suchthat(L)isunchangedwhenanylocalknotisaddedtoL.Since=(),clearlyBissuchalink. Inparticular,sinceKLtype1invariantsofsingularlinksareinvariantundertyinglocalknotsduetotheoneterm(framingindependence)relation,itfollowsthatG2()=~G2()im: Proof. SincetheinclusionP0[f(J0),!P1isnullhomotopic,thereexistsagenericPLhomotopyf0t:f(mS1)!S3suchthatthehomotopyft:=f0tf:mS1!S3satisesf0=f,ft(J0)P1,ft=foutsideJ0,andf1(~J0)isasmallcircle,boundinganembeddeddiskinthecomplementtof1(mS1).Usingtheoneterm(framingindependence)relation,weseethatanyniteKLtypeinvariantvanishesonf1.HenceanyKLtype1vanishesonf0=f,whichcompletestheprooffork=1. Beforeproceedingtothegeneralcase,westatethefollowinggeneralizationoftheoneterm(framingindependence)relation. Proof.
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thedenitionofkquasiisotopy,k>2,wewillnowconstructagenericPLhomotopyf0t1;t:ft1(mS1)!S3suchthatthehomotopyft1;t:=f0t1;tf:mS1!S3satisesft1;0=ft1,ft1;t(J1)P2,ft1;t=ft1outsideJ1,andft1;1takesJ1intoaballB1inthecomplementtoft1;1(mS1nJ1). Indeed,sinceft1(J1)P1[f(J1)isnullhomotopicinP2,wecancontractthe1dimensionalpolyhedronft1(J1nU),whereUisaregularneighborhoodof@J1inJ1,embeddedbyft1,intoasmallballB01S3nft1(mS1nJ1),byahomotopyft1;tjJ1asrequiredabove.Joiningtheendpointsofft1(mS1n@J1)tothoseofft1;t(J1nU)bytwoembeddedarcsft1;t(U)inanarbitraryway(continuouslydependingont),weobtaintherequiredhomotopy.TakingasmallregularneighborhoodofB01[ft1;1(J1)relativetoft1;1(@J1)weobtaintherequiredballB1. (i)fti1;:::;tin;t=fti1;:::;tinfort=0andoutsideJn; (ii)fti1;:::;tin;t(Jn)Pn+1foreacht; (iii)f1ti1;:::;tin;1(Bn1)=JnforsomePL3ballBn1. Remark.Infact,theproofofTheorem2.2worksunderaweakerassumptionandwithastrongerconclusion.Namely,kquasiisotopycanbereplacedwithvirtualkquasiisotopy(seedenitioninMelikhovandRepovs[ 25 ]),whereasindistinguishabilitybyKLtypek 25 ],thehomotopyft1;tcanbevisualizedasshiftingthearcsf(I0j)ontothearcsF(Ij)andthentakingtheimageofJ1intoaballalongthetrackofthenullhomotopyF.
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invariants(whichcanbethoughtofasaformalgeneralizationofkequivalenceinthesenseofGoussarov[ 10 ],HabiroandStanford)canbereplacedwithgeometrickequivalence,denedasfollows.Foreachn>0,letLMn;0denotethesubspaceofLMnconsistingofthelinkmapslsuchthatallsingularitiesoflarecontainedinaballBsuchthatl1(B)isanarc.Wecalllinkmapsl;l02LMngeometricallykequivalentiftheyarehomotopicinthespaceLMn[LMn+1;k,whereLMi;kfork>0,i>0consistsofthoselinkmapswithisingularitieswhicharegeometrically(k1)equivalenttoalinkmapinLMi;0. Toseewhygeometrickequivalencemaydierfromkquasiisotopy,weemploythenotionofweakkquasiisotopy,whichisdenedsimilarlytokquasiisotopy,butwithnullhomotopiesreplacedbynullhomologies,cf.[ 26 ]. Thedierencebetweengeometrickequivalenceandweakkquasiisotopyisanalogoustothedierencebetweenthelowercentralseriesandthederivedseries,andfork>2canbevisualizedasfollows.LetMkdenotethekthMilnorlink(cf.,e.g.,Fig.1inMelikhovandRepovs[ 26 ]),andletMWk,k>2,beMkwherethe\long"componentisreplacedbyitsWhiteheaddouble.Unclaspingtheclaspofthedoubledcomponentisa1quasiisotopyhtfromMWktotheunlink.Thishtisnota2quasiisotopy(evennotaweak2quasiisotopy),sinceMkisnotaboundarylink,asdetectedbyCochran'sinvariants(cf.,e.g.,[ 26 ]).ItwouldthusbeinthespiritofConjecture1.1inMelikhovandRepovs[ 26 ]toconjecturethatthereexistsno2quasiisotopy,evenweak,fromMWktotheunlink.However,sinceMkis(k1)quasiisotopictotheunlink,htrealizesgeometrickequivalencebetweenMWkandtheunlink. Fork=1nosuchexampleexists,sincegeometric1equivalenceclearlyimpliesweak1quasiisotopy.For2componentlinkstheconversealsoholds(cf.Fig.2inMilnor[ 27 ]),amplifyingtheanalogywiththelowercentralseriesandthederivedseries,whoserespectiveinitialtermscoincide.Thusthedierencebetween1quasiisotopyandgeometric1equivalencefor2componentlinksreducestothedierencebetween1quasiisotopyandweak1quasiisotopy(seeFig.2(d)inMelikhovandRepovs[ 25 ]). Existenceofdierencebetweengeometrickequivalenceand\indistinguishabilitybyKLtypekinvariants,welldeneduptoPLisotopy,"alreadyfork=1,wouldfollow
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fromthe(expected)positivesolutiontoProblem1.5inMelikhovandRepovs[ 25 ]forweak1quasiisotopy.
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RecallthattheConwaypolynomialofalinkListheuniquepolynomialrL(z)satisfyingrunknot=1andthecrossingchangeformularL+(z)rL(z)=zrL0(z); whereL+andLdierbyapositivecrossingchange,andL0isobtainedbyorientedsmoothingoftheselfintersectionoftheintermediatesingularlinkLs.NotethatL0hasonemore(respectivelyless)componentthanL+andLiftheintersectionisaselfintersectionofsomecomponent(resp.anintersectionbetweendistinctcomponents). Theskeinrelation(1)showsthatthecoecientatzkinrLisatypekinvariant.ThegeneralizedonetermrelationfornitetypeinvariantsspecializesheretotheequationrL=0foranysplitlinkL(i.e.,alinkwhosecomponentscanbesplitintwononemptypartsbyanembeddedsphere).Using(1)itisnoweasytoseethattheConwaypolynomialofanmcomponentlinkLisnecessarilyoftheformrL(z)=zm1(c0+c1z2++cnz2n) forsomen.By(1),c0(K)=1foranyknotK,whichcanbeusedtorecursivelyevaluatec0onanylinkL.Forexample,itisimmediatethatc0(L)isthelinkingnumberofLform=2,andc0(L)=ab+bc+caform=3,wherea;b;carethelinkingnumbersofthe2componentsublinks(cf.[ 22 ]).Forarbitrarym,itiseasytoseethatc0(L)isasymmetricpolynomialinthepairwiselinkingnumbers(andthusaKLtype0invariant);see[ 18 23 ]foranexplicitformula. Proof. 10
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accordingascisontoornot;ineithercase,itisactype0invariant.Aninductiononiusingtheskeinrelation(1)showsthatthecoecientofrLatzjcj1+iisofctypei.Inthecasewherecistheidentity,thisisourassertion. InordertocomputerKforaknotK,wecouldconsiderasequenceofcrossingchangesinaplanediagramofK,turningKintotheunknot(cf.[ 31 ]).ThenrK=runknot+zP"irLi,whereLiarethe2componentlinksobtainedbysmoothingthecrossingsand"i=1arethesignsofthecrossingchanges.WecouldfurtherconsiderasequenceofcrossingchangesinthediagramofeachLi,involvingonlycrossingsofdistinctcomponentsandturningLiintoasplitlink.ThisyieldsrLi=zP"ijrKij,whereKijaretheknotsobtainedbysmoothingthecrossingsand"ij=1arethesignsofthecrossingchanges.SincethediagramofeachKijhasfewercrossingsthanthatofK,wecanexpressrK,iteratingthisprocedure,asPkz2k(Pikirunknot),wherethesignski=1aredeterminedbytheaboveconstruction.Sincerunknot=1,plainlyrK=Pkz2kPiki.NotethatthisprocedureshowsthatrKisindeedapolynomial(ratherthanapowerseries)foranyknotK;asimilarargumentworksforlinks. NowletLbealink,andsupposethatL0isobtainedfromLbytyingaknotKlocallyononeofthecomponents.Wecanechotheaboveconstruction,expressingrL0asPkz2kPikirL,wherethesignskiaresameasabove.Thus 20 ],butweneedtheargumentinordertosetupnotationforuseintheproofofTheorem3.4.
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wherei=ci(i1b1++0bi).HencerL2Z[[z]](ratherthanjustQ[[z1]]),andckk(modgcd(c0;:::;ck1)).Inparticular,form=2weseethat0(L)=c0(L)isthelinkingnumber,and1(L)=c1(L)c0(L)(c1(K1)+c1(K2))is(cf.[ 22 ])thegeneralizedSato{Levineinvariant! UnderthegeneralizedSato{LevineinvariantwemeantheinvariantthatemergedintheworkofPolyak{Viro(see[ 2 ]),Kirk{Livingston[ 16 22 ],Akhmetiev(see[ 1 ])andNakanishi{Ohyama[ 30 ];seealso[ 25 ]. 16 ]).Thisisnotsurprising,becausec1hasKLtype2byProposition3.1,hence(c1)isinvariantunderPLisotopybyRemark(ii)inx2. (iii)TheextendedrLisinvariantunderTOPisotopyofL.
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assertions(i)and(ii)nowfollowfromTheorem1.3(a),and(iii)eitherfrom[ 25 ]or,alternatively,from(ii)andcompactnessoftheunitinterval. Proposition3.1impliesthateachk(L)isofKLtype2k.Itiseasytocheckthat1(L)=c1(L)c0(L)(Pic1(Ki))is,infact,ofKLtype1andthat2(L)=c2(L)1(L)Xic1(Ki)!c0(L)Xic2(Ki)+Xi6=jc1(Ki)c1(Kj)! Proof. where,asusual,+andstandfortheoverpassandtheunderpass,and0forthesmoothingofthecrossings.AssumingthateachLontherighthandsidehasthreecomponentsandeachKonlyonecomponent,wecansimplifythisasc0(L000)c0(L++0)c0(L0+0)c0(L00+). Let':S1#S2beagenericC1approximationwith3doublepointsoftheclockwisedoublecoverS1!S1R2,andletA;B;C;Ddenotethe4boundedcomponentsofR2n'(S1)suchthatDcontainstheorigin.LetKsss2LM13bethecompositionof'andtheinclusionR2S2S3,andletK:S1,!S3nKsss(S1)beaknotinthecomplementofKssslinkingtheclockwiseorientedboundariesofA;B;C;Dwithlinkingnumbersa;b;c;dsuchthata+b+c+2d=0.Finally,deneLsss2LM23tobetheunionofKandKsss,thenlk(Lsss)=a+b+c+2d=0andwendthat2(Lsss)=(a+b+c+d)d(a+c+d)(b+d)(a+b+d)(c+d)(b+c+d)(a+d):
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Thisexpressionisnonzero,e.g.,fora=1,b=2,c=3,d=0. ByProposition3.1andTheorem2.2,n(L)isinvariantunder2nquasiisotopy.However,accordingtoProposition3.3,thefollowingstrengtheningofthisassertioncannotbeobtainedbymeansofTheorem2.2. NowletLbeanmcomponentlink,andsupposethatL0isobtainedfromLbyagenericPLlinkhomotopyHwithasinglesingularlevelLs.ColorthecomponentsofLwithdistinctcolors,thenthe(m+1)componentsmoothingL0ofLsisnaturallycoloredwithmcolors.SupposeinductivelythatL0i2:::ik+1isoneofthesmoothedsingularlinks
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inagenericPLcoloredlinkhomotopyH0i2:::ikbetweenthelinkL0i2:::ikandsomelinkL00i2:::ikwhicheitherhasonelesscomponentthanL0i2:::ikorcoincides(geometrically)withL.IfthereexistsuchhomotopiesH0i2:::ilforl=1;:::;2k,wesaythatLsisa+()skeinkquasiembedding,providedthattheselfintersectioninHispositive(negative).Askeinkquasiisotopyisalinkhomotopywhereeverysingularlevelisa+orskeinkquasiembedding,dependingonthesignoftheselfintersection. Toseethatnisinvariantunderskeinnquasiisotopy Tocompletetheproof,weshowthatnquasiisotopyimpliesskeinnquasiisotopy.ItsucestoconsideracrossingchangeonacomponentKofL,satisfyingthedenitionofnquasiisotopy.Letf,J0;:::;JnandP0;:::;Pnbeasinthedenitionofnquasiembedding;wecanassumethatP1containsaregularneighborhoodoff(J0)containingL(J0).WeassociatetoeverylinkL0,suchthatLnKisageometricsublinkofL0,butLitselfisnot,thecollectionofpositiveintegersd(L0)=(d0;:::;dm),wheredifori>0(resp.i=0)istheminimalnumbersuchthattheithcomponentofL0notinLnKisnullhomotopicinPdi(resp.ishomotopictoKwithsupportinPdi).Itiseasytoseethat 2)quasiisotopy.
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Supposethatnoneoftheintegersd(L0i2:::ik)exceedsl+1,andatleastrofthemdonotexceedl.Ifr>1,letdibeoneoftheser,thennoneoftheintegers(d(L0i2:::ik)),whereisiorij,exceedsl+1,andatleastr1ofthemdonotexceedl.ThusifatleasttwocomponentsofL0i2:::ikarenotinLnKandnoneoftheintegersd(L0i2:::ik)exceedsl,wecanconstructahomotopyH0i2:::ikasabove,andforeachsingularlinkL0i2:::ik+1inthishomotopyalsoahomotopyH0i2:::ik+1asabove,sothatforeachsingularlinkL0i2:::ik+2inthishomotopy,noneoftheintegersd(L0i2:::ik+2)exceedsl+1.Butitisindeedthecasefork=1,henceforanyoddk,thatatleasttwocomponentsofL0i2:::ikarenotinLnK. Sinceckk(modgcd(c0;:::;ck1)),Theorem3.4impliesthecasel=1,andalsothe2componentcaseof 26 ]Set=d(l1)(m1) 2e.Theresidueclassofc+kmodulogcdofc;:::;c+k1andallinvariantsoflength6lisinvariantunder(b m1c+k)quasiisotopy. (Heredxe=nifx2[n1 2;n+1 2),andbxc=nifx2(n1 2;n+1 2]forn2Z.)
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befoundamongcoecientsofrationalfunctionsinrofthelinkanditstwocomponentsublinks.Itturnsoutthatthisisnotthecasealreadyfork=1.Indeed,wewouldgetanintegerlinkhomotopyinvariantof(monochromatic)nitetype(specically,oftype4),whichisnotafunctionofthepairwiselinkingnumbers(sincec1(123)2(mod1),cf.[ 18 23 ]).Butthisisimpossiblefor3componentlinks[ 24 ]. Alternatively,onecanarguedirectlyasfollows.Considertheinvariant(L):=1(L)P0(L0)1(L1),wherethesummationisoverallorderedpairs(L0;L1)ofdistinct2componentsublinksofL.(Recallthatinthe2componentcase0(L)and1(L)coincidewiththelinkingnumberandthegeneralizedSatoLevineinvariant,respectively.)Itcanbeeasilyveriedthat(L)jumpsby(12)(1;3+)(2;3)+(1;3)(2;3+) 1andr(123)2(mod1),06r<1.
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TheConwaypolynomialisequivalenttothemonochromaticcaseoftheConwaypotentialfunctionLofthecoloredlinkL,namelyrL(xx1)=(xx1)L(x)formonochromaticL.ForalinkLcoloredwithncolors,L2Z[x11;:::;x1n](theringofLaurentpolynomials)ifLhasmorethanonecomponent;otherwiseLbelongstothefractionalideal(xx1)1Z[x1]intheeldoffractionsofZ[x1]. L(x1;:::;xn)isanormalizedversionofthesignrenedAlexanderpolynomialL(t1;:::;tn),whichiswelldeneduptomultiplicationbymonomialsti11:::tinn.IfKisaknot,(xx1)K(x)=xK(x2),whereasforalinkLwithm>1components,L(x1;:::;xn)=x11:::xnnL(x2c(1);:::;x2c(m)),wherec:f1;:::;mgf1;:::;ngisthecoloring,andtheintegers;1;:::;nareuniquelydeterminedbythesymmetryrelationL(x1;:::;xn)=(1)mL(x11;:::;x1n).WerefertoHartley[ 11 ],Traldi[ 35 ]andTuraev[ 36 ]fordenitionofthesignrenedAlexanderpolynomialanddiscussionoftheConwaypotentialfunction. arisesgeometricallyfromconsideringliftsofthegeneratorsofthefundamentalgroupofthelinkcomplementtotheuniversalabeliancover,startingataxedlift~pofthebasepoint.Ifwewishtobasethewholetheoryontheliftsendingat~p,wewillhavetochangetheLeibnizruletoD(fg)=D(f)g1+D(g):
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The\symmetric"LeibnizruleD(fg)=D(f)g1+fD(g)+(f1g1g1f1) doesnotseemtohaveacleargeometricmeaning.(Notetheanalogywiththethreerstordernitedierences.)However,itdoescorrespondtoa\symmetric"versionoftheMagnusexpansion,whichwillbebehindthescenesinthissection.Indeed,itsabelianversiongivesrise,inTheorem4.2below,toapowerseriesfL,inthesamewayastheabelianversionoftheusualMagnusexpansionleadstoTraldi'sparametrizationofL,discussedintheproofofTheorem4.8below. 15 ])(a)L(x1;:::;xn)=L(x11;:::;x1n). (b)IfLhasm>1components,thetotaldegreeofeverynonzerotermofLiscongruenttom(mod2). (c)IfLhas>1components,thexidegreeofeverynonzerotermofLiscongruentmod2tothenumberkiofcomponentsofcoloriplusli:=Plk(K;K0),whereKrunsoverallcomponentsofcoloriandK0overthoseofothercolors. Proof. 35 ]).Thisproves(c),whichimplies(b).Forlinkswith>1components(a)followsfrom(b)andfromthesymmetryrelationinthedenitionofL;forknots{fromtherelationofL(z)withtheConwaypolynomial. ComputationofLismuchharderthanthatofrL,fortheskeinrelation(1)isnolongervalidforarbitrarycrossingchanges.Itsurvives,intheformL+L=(xix1i)L0; inthecasewherebothstrandsinvolvedintheintersectionareofcolori.Thereareotherformulasforpotentialfunctionsoflinksrelatedbylocalmoves,whichsuceforevaluationofonalllinks;seereferencesinMurakami[ 29 ]. Dynnikov[ 6 ]andH.Murakami[ 29 ]noticedthatthecoecientsofthepowerseriesL(eh1=2;:::;ehn=2)are(monochromatic)nitetypeinvariantsofL.
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2t2+:::.TheGaloisgroupofthequadraticequationz=xx1actsonitsrootsbyx7!x1,soLemma4.1(a)impliesthatthecoecientsofthepowerseriesfL(z1;:::;zn):=L(x(z1);:::;x(zn)) areindependentofthechoiceoftherootx. ThesecondassertionfollowsbyMurakami'sproof[ 29 ]oftheresultmentionedabove,andthethirdfromtheskeinrelation(2).Next,sincex1x=(xx1),fL(z1;:::;zn)=L(x(z1)1;:::;x(zn)1)=(1)mL(x(z1);:::;x(zn))=(1)mfL(z1;:::;zn); Thenalassertionmaybenotobvious(since(k+1)2k1kfork=1;3;7)from(1+4y)1=2=1+4y1Xk=0(1)k
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However,thispowerseriesdoeshaveintegercoecients,sincethisisthecasefor(1+4y)1=2=1+21Xk=1(1)k2k1kyk: Letusconsiderthecaseoftwocolorsinmoredetail.LetcijdenotethecoecientoffL(z1;z2)atzi1zj2.Sincef(z;z)=z1r(z),c00coincideswiththelinkingnumberc0inthe2componentcaseandiszerootherwise.Theskeinrelation(2)impliesthatfora3componentlinkK1[K01[K2,coloredasindicatedbysubscript,c10coincideswithlk(K1;K01)(lk(K1;K2)+lk(K01;K2))uptoatype0invariantofthecoloredlink.Thelatterisidenticallyzerobytheconnectedsum whichfollowsfromthedenitionofLinHartley[ 11 ].Sincec01=c0c10orbyananalogousargument,c01=lk(K1;K2)lk(K01;K2).Itfollowsthatfor2componentlinksc11coincideswiththegeneralizedSato{Levineinvariant1uptoaKLtype0invariant.Since1c11isoftype3,ithastobeadegree3polynomialinthelinkingnumber,whichturnsouttobe1 12(lk3lk).(Sinceunlink=0andHopflink=1,thesecondConwayidentity[ 11 15 ]canbeusedtoevaluateLonaseriesoflinksHnwithlkHn=n.)Thehalfintegerc11isthustheunorientedgeneralizedSato{Levineinvariant,whichisimplicitinthesecondparagraphofKirkandLivingston[ 16 ].Specically,2c11=lk2is(unlesslk=0)theCasson{WalkerinvariantoftheQhomologysphereobtainedby0surgeryonthecomponentsofthelink;infact,c11(L)=1(L)+1(L0) 2,whereL0denotesthelinkobtainedfromLbyreversingtheorientationofoneofthecomponents[ 16 ]. Theprecedingparagraphimpliestherstpartofthefollowing
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(i)00=lk,and11istheunorientedgeneralizedSato{Levineinvariant,whichassumesallhalfintegervalues; (ii)1;2k1and2k1;1coincide,whenmultipliedby(1)k+1,withCochran's[ 5 ]derivedinvariantsk,wheneverthelatteraredened(i.e.,lk=0); (iii)ijisoftypei+j;wheni+jisodd,ij=0. 5 ]thatP1i=1iL(z2)i=L(y2),whereLisKojima'sfunctionandz=yy1.Ontheotherhand,itfollowsfrom[ 13 ]thatL(y2)=(1y2)(1y2)0L(1;y) 13 ](seealsoproofofTheorem4.8(ii)below)thatunderthisassumption0LisaLaurentpolynomial(ratherthanjustarationalfunction).Denef0LbyfL(z1;z2)=z1z2f0L(z1;z2)for2componentlinkswithlk=0,then1Xi=1(1)iiLz2i=z2f0L(0;z) ItturnsoutthateachcoecientofthepowerseriesfcanbecanonicallysplitintoaQlinearcombinationofthecoecientsofcertain2n1polynomials.
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(4) 2Z[z1;:::;zn],whereff(x1;:::;xn)gdenotesf(x1;:::;xn)+f(x11;:::;x1n)foranyfunctionf(x1;:::;xn).Moreover,ifn=2l,thecoecientsofP12:::nareinteger. 15 ]:(x;y)=K1(xx1;yy1)+(xy+x1y1)K2(xx1;yy1): whereMisanymonomialinx11;:::;x1n,canbeverieddirectly(separatelyforMofoddandeventotaldegrees),andallowstoexpresseachfxp11:::xpnngintheformX16i1<
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forsomeP0i1:::ik2Z[z1;:::;zn],k>0,andsomeP021 2Z[z1;:::;zn].Thesummandscorrespondingtok=1canbeincludedinP0,andonecangetridofthesummandscorrespondingtooddk>3byrepeateduseoftheformula2fxix1jMg=fxigfx1jMg+fx1jgfxiMg+fxixjgfMg; combinedwith().Notethat()followsimmediatelyfrom()anditsanaloguefxixjMg+fx1ix1jMg=fxixjgfMg: 2P0j1:::j2k. Itremainstoverifyuniquenessof(4).Suppose,bywayofcontradiction,thatanontrivialexpressionQ(x1;:::;xn)intheformoftherighthandsideof(4)isidenticallyzero.ThensoisQ(x1;:::;xn1;xn)Q(x1;:::;xn1;x1n),whichcanberewrittenas[xn]X16i1<
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Repeatingthistwostepprocedurebn accordingasnisevenorodd.Consider,forexample,thecaseofoddn.Bysymmetry,P1:::^{:::n=0foreachi.Returningtothepreviousstage[x4]X16i1<
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Moregenerally,wehave Proof.
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LetK1;:::;KmdenotethecomponentsofacoloredlinkL,wherem>1.ItfollowsfromTheorem4.4thatthepowerseriesL(x1;:::;xn):=L(x1;:::;xn) canbeexpressedintheformL(x1;:::;xn)=X16i1<
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compare[ 35 32 ].Thereis,ofcourse,suchasplittingforeveryPi1:::i2n,producingaplethoraofPLisotopyinvariants. LackofananalogueofLemma4.5(ii)forthepowerseriesfLhindersestablishingasimplerelationbetweenthecoecientsoffLandthoseofTraldi'sexpansionofL[ 35 ],whichpreventsonefromexpressingthenitetyperationalsij(discussedinTheorem4.3)asliftingsofinvariants.However,eachijcanbesplitintoalinearcombination(whosecoecientsdependonthelinkingnumber)oftheintegercoecientsofthepowerseries5L,whichdoadmitsuchanexpression. (i)00=lk,and11isanintegerliftingoftheSato{Levineinvariant(1122),butnotanitetypeinvariant; (ii)1;2k1and2k1;1areCochran's[ 5 ]derivedinvariants(1)k+1k,wheneverthelatteraredened(i.e.,lk=0); (iii)ijisanintegerliftingofMilnor's[ 28 ]invariant(1)j(1:::1 {z }i+12:::2 {z }j+1),providedthati+jiseven; (iv)ijisofKLtypei+j; (v)wheni+jisodd,ij=0; (vi)wheni+1,j+1andlkareallevenorallodd,ijiseven; (vii)foragivenL,thereareonlynitelymanypairs(i;j)suchthatij60modulothegreatestcommondivisorijofallkl'swithk6i,l6jandk+l
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whereLdenotestheusualAlexanderpolynomial,`theAlexanderpolynomialofthestringlink[ 19 ],i.e.,theReidemeistertorsionofthebasedchaincomplexofthepair(X;X\R2f0g),whereX=R2Inim`,and`acertainrationalpowerseries,determinedbytheinvariantsof`. forsomeQ1;Q22Z[z1;z2],where=0or1accordingaslk(L)isoddoreven.Setyi=x2i1,thenx2i=1yi+y2iy3i+:::.Theidentitiesfxig2=x2i(x2i1)2=(1yi+y2i:::)y2i;fx1x12gfx1gfx2g=(x21+x22)(x211)(x221)=(2y1y2+y21+y22:::)y1y2 Letusstudythissubstitutionmorecarefully.Letd0ij,d00ijdenotethecoecientsatzi1zj2inPandP12,andletuswrite(k;l)6(i;j)ifk6iandl6j.Thenthecoeciente00ijatyi1yj2inthepowerseriesR2,denedbytheequality(y1y2)1R2(y1;y2)=(z1z2)1P12(z1;z2),isgivenbye00ij=X(k;l)6(i;j)(1)(ik)+(jl)d00kl=(1)i+jX(k;l)6(i;j)d00kl
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Thekeyobservationhereisthatallcoecientsontherighthandsideareevenifi+jiseven. Letusconsiderthecase=1.ThenbyLemma4.5(ii),d00ij=0unlessbothiandjareeven,andd0ij=0unlessbothiandjareodd.Hencee00ij0modulogcdfe00klj(k;l)<(i;j)g,unlessbothiandjareeven,andsimilarlyfore0ij.Nowitfollowsbyinductionthateij2e00ij2d00ijoreij2e0ij2d0ijmoduloEij:=gcdfeklj(k;l)<(i;j)gaccordingasiandjarebothevenorbothodd.Thuseij2(d0ij+d00ij)(modEij)ifi+jiseven.Clearly,thelatterassertionholdsinthecase=0aswell,whichcanbeprovedbythesameargument. Finally,sincelk1(mod2),andx2iareexpressibleaspowerseriesinyiwithintegercoecientsandconstantterm1,thecoecients~eijofTraldi'spowerseries~TL(y1;y2)=(x1x2)lk1LarerelatedtoeijbycongruencemodEij.Thiscompletestheproof,sincebyTraldi[ 35 ],each~eijisanintegerliftingof(1)j+1(1:::1 {z }i+12:::2 {z }j+1). {z }i+12:::2 {z }j+1)=0wheni+jisodd[ 34 ]. 28 ].Henceby(iii),0;2k2k;00(modlk)foreachk.Soiflk=0,everynonzerotermofeitherPorP12involvesbothz1andz2.(Alternatively,thisfollowsfromJin'slemmamentionedintheproofofTheorem4.3(ii).)ByLemma4.5(ii),everynonzerotermofP12hastofurtherincludeeachofthemonceagain,i.e.,P12isdivisiblebyz21z22.Hence,rstly,2k1;1coincideswiththecoecientof2P(z1;z2)atz2k11z2,and,secondly,thiscoecientisnotaectedbyadding(x(z1)x(z2)1+x(z2)x(z1)1)P12(z1;z2)to2P(z1;z2),wherex(z)isasintheproofofTheorem4.2. Proof.
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26 ]Cochran'sinvariantskareinvariantunderkquasiisotopy; (b)Milnor'sinvariants(1:::12:::2)ofevenlengthareinvariantunderkquasiisotopy,ifeachindexoccursatmost2k+1times. 26 ]Allinvariantsoflength62k+3areinvariantunderkquasiisotopy.
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WerecallthattheHOMFLY(PT)polynomialandtheDubrovnikversionoftheKaumanpolynomialaretheuniqueLaurentpolynomialsHL;FL2Z[x1;y1]satisfyingHunknot=Funknot=1andxHL+x1HL=yHL0;xFL+x1FL=y(FL0xw(L1)w(L0)FL1); 20 ]areobtainedasHL(ia;iz)and(1)m1FL(ia;iz). (i)ForeachL:mS1,!S3andanyn2Nthereexistsan"n>0suchthatifL0:mS1,!S3isC0"nclosetoL,HL0(ech=2;eh=2eh=2)HL(ech=2;eh=2eh=2)mod(hn);FL0(e(c1)h=2;eh=2eh=2)FL(e(c1)h=2;eh=2eh=2)mod(hn): (iii)TheextendedHLandFLareinvariantunderTOPisotopyofL.
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20 ]implythatHandFareinvariantunderPLisotopy.(NotethattheconnectedsuminLickorish[ 20 ]isHashizume's,notthecomponentwiseconnectedsumofMelikhovandRepovs[ 25 ].)Ontheotherhand,itwasnoticedinLieberum[ 21 ](compare[ 9 4 33 ])thatthecoecientsofthepowerseriesHL(ech=2;eh=2eh=2)=1Xk=0k+m1Xi=0pkicihk2Q[c][[h]]FL(e(c1)h=2;eh=2eh=2)=1Xk=0k+m1Xi=0qkicihk2Q[c][[h]] are(monochromatic)nitetypeinvariantsofL.Specically,eachpkiandeachqkiisoftypek,moreoverp0i=q0i=m1;i(theKroneckerdelta).(TheargumentinLieberum[ 21 ]wasforHLHT2andFLFT2,whereT2denotesthetrivial2componentlink,butitworksaswellforHLandFL,compare[ 3 ].)TherestoftheproofrepeatsthatofTheorem3.2.
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IwasborninMoscow,Russia,in1980.MyhighschoolwastheMoscowStateFiftySeventhSchool,whereIstudiedinamathematicallyorientedclassin1992{96.In1996{2001IcontinuedmyeducationattheMechanicsandMathematicsDepartmentoftheMoscowStateUniversity,whichconferredabachelor'sdegreeinpureandappliedmathematicsonme.IhavebeenintheGraduateSchooloftheUniversityofFloridasince2001. 37
