## Milnor invariants and the HOMFLYPT Polynomial

JEAN-BAPTISTEMEILHAN

AKIRA YASUHARA

We give formulas expressing Milnor invariants of ann–component linkL in the 3–sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invari- antxL.J/vanishes for any sequence J with length at most k, then any Milnor x–invariant xL.I/ with length between 3 and 2kC1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the “first nonvanishing” Milnor invariants can be always represented as such a linear combination.

57M25, 57M27

## 1 Introduction

J Milnor defined in [19; 20] a family of link invariants, known asMilnorx–invariants.

Here, and throughout the paper, by a link we mean an oriented, ordered link in S^{3}.
Roughly speaking, Milnor invariants encode the behaviour of parallel copies of each
link component in the lower central series of the link group. Given an n–component
link L in S^{3}, Milnor invariants are specified by a sequenceI of (possibly repeating)
indices from f1; : : : ;ng. The length of the sequence is called thelengthof the Milnor
invariant. It is known that Milnor invariants of length two are just linking numbers.

However in general, Milnor invariant xL.I/ is only well-defined modulo the greatest common divisorL.I/of all Milnor invariantsxL.J/such thatJ is obtained fromI by removing at least one index and permuting the remaining indices cyclicly. This indeterminacy comes from the choice of the meridian curves generating the link group.

Equivalently, it comes from the indeterminacy of representing the link as the closure of a string link; see Habegger and Lin [7]. See Section 4 for the definitions.

Recall that the HOMFLYPT polynomial of a knot K is of the form P.KIt;z/D PN

kD0P2k.KIt/z^{2k}, and denote byP_{0}^{.}^{l}^{/}.K/thel–th derivative ofP0.KIt/2ZŒt^{˙}^{1}
evaluated at t D1. Denote by .logP0.K//^{.}^{n}^{/} the n–th derivative of logP0.KIt/
evaluated at t D1. Since P_{0}.KI1/D1 andP_{0}^{.}^{1}^{/}.K/D0, we have

.logP0/^{.}^{n}^{/}DP_{0}^{.}^{n}^{/}C X

k_{1}CCkmDn

n_{.}k_{1};:::;k_{m}/P_{0}^{.}^{k}^{1}^{/}: : :P_{0}^{.}^{k}^{m}^{/};

where the sum runs over allk1; : : : ;kmsuch thatk1C CkmDn.ki2/, and where
n_{.}_{k}_{1}_{;:::;}_{k}_{m}_{/}2Z. For example, one can check that .logP_{0}/^{.}^{n}^{/}DP_{0}^{.}^{n}^{/} for nD1;2;3,
and that .logP_{0}/^{.}^{4}^{/}DP_{0}^{.}^{4}^{/} 3.P_{0}^{.}^{2}^{/}/^{2}.

In this paper, we show the following. If Milnor invariant xL.J/ vanishes for any
sequence J with length at most k, then any Milnor x–invariant xL.I/ with length
mC1.3mC12kC1/ is given by a linear combination of.logP_{0}/^{.}^{m}^{/} invariants
of knots obtained from the link by certain band sum operations.

For simplicity, we first state the formula for Milnor link-homotopy invariants .x I/, ie such that the sequence I has no repeated index. Let L DSn

iD1Li be an n–

component link in S^{3}. LetI Di1i2: : :im be a sequence of m distinct elements of
f1; : : : ;ng. Let B_{I} be an oriented .2m/–gon, and denote by pj .j D1; : : : ;m/ a
set of m nonadjacent edges of B_{I} according to the boundary orientation. Suppose
that B_{I} is embedded in S^{3} such that B_{I} \L DSm

jD1pj, and such that each pj

is contained in Li_{j} with opposite orientation. We call such a disk an I–fusion disk
for L. For any subsequence J of I, we define the oriented knot L_{J} as the closure of

.S

i2fJgLi/[@BI n .S

i2fJgLi/\BI

, wherefJg is the subset off1; : : : ;ngof all indices appearing in the sequence J.

Theorem 1.1 LetL be ann–component link in S^{3} (n3) with vanishing Milnor
link-homotopy invariants of length up to k. Then for any sequence I of .mC1/
distinct elements of f1; : : : ;ng .3mC12kC1/and for anyI–fusion disk forL,
we have

xL.I/ 1
m!2^{m}

X

J<I

. 1/^{j}^{J}^{j}.logP_{0}.L_{J}//^{.}^{m}^{/} .modL.I///;

where the sum runs over all subsequences J ofI, and wherejJjdenotes the length of the sequence J.

This generalizes widely a result of M Polyak for Milnor’s triple linking number .x 123/ [22]. There are several other known results relating Milnor invariants of (string) links to the Alexander polynomial; for example see Cochran [3], Levine [13], Masbaum and Vaintrob [16], Meilhan [17], Murasugi [21] and Traldi [24; 25] (note in particular that the results of [13; 16; 17; 22] make use of closure-type operations).

The relationship to quantum invariants is also known via the Kontsevich integral; see Habegger and Masbaum [8].

We emphasize that our assumption that the link has vanishing Milnor link-homotopy invariants of length up to k is essential in order to compute its Milnor invariants of length up to 2kC1 using our formula. See the example at the end of this paper, which

shows that Milnor link-homotopy invariants of length 4 are not given by the formula in Theorem 1.1 if there are nonvanishing linking numbers.

As noted above, only the first nonvanishing Milnor invariants of a link are well-defined integer-valued invariants. The following is in some sense a refinement of Theorem 1.1 for the first nonvanishing Milnor invariants.

Theorem 1.2 Let L be ann–component link in S^{3} (n3) with vanishing Milnor
link-homotopy invariants of length up tok.2/. Then for any sequenceI of .kC1/
distinct elements of f1; : : : ;ngand for any I–fusion disk forL, we have

xL.I/D 1
k!2^{k}

X

J<I

. 1/^{j}^{J}^{j}P_{0}^{.}^{k}^{/}.L_{J}/2Z:

Theorem 1.2 implies that all Milnor link-homotopy invariants of a linkL vanish if and only if all linking numbers of L vanish and P

J<I. 1/^{j}^{J}^{j}P_{0}^{.}^{k}^{/}.LJ/D0 for all k
.2kn 1/ and for all nonrepeated sequences I of length kC1.

We remark that since the HOMFLYPT polynomial of knots is preserved by mutation, by Theorem 1.2, the first nonvanishing Milnor link-homotopy invariants are also preserved (cf Cha [2, Theorems 1.4 and 1.5]).

Theorem 1.1 and Theorem 1.2 generalize as follows. Let L DSn

iD1Li be an n–

component link in S^{3}, and let I D i_{1}i_{2}: : :im be a sequence of m elements of
f1; : : : ;ng, where each element i appears exactly ri times. Denote by D_{I}.L/ the
m–component link obtained fromL as follows.

Replace each string Li by ri zero-framed parallel copies of it, labeled from
L_{.}_{i}_{;}_{1}_{/} toL_{.}_{i}_{;}_{r}_{i}_{/}. If ri D0 for some indexi, simply delete Li.

Let D_{I}.L/DL^{0}_{1}[ [L^{0}_{m} be the m–string link S

i;jL_{.}_{i}_{;}_{j}_{/} with the order
induced by the lexicographic order of the index.i;j/. This ordering defines a
bijection 'W f.i;j/j1i n;1jrig ! f1; : : : ;mg.

We also define a sequence D.I/of elements of f1; : : : ;mgwithout repeated index as follows. First, consider a sequence of elements off.i;j/j1i n;1jrig by replacing each i in I with .i;1/; : : : ; .i;ri/ in this order. For example if ID12231, we obtain the sequence.1;1/; .2;1/; .2;2/; .3;1/; .1;2/. Next replace each term.i;j/ of this sequence with '..i;j//. Hence we have D.12231/D13452.

Theorem 1.3 LetL be ann–component link inS^{3} with vanishing Milnor invariants
of length up to k. Let I be a sequence of .mC1/ possibly repeating elements of
f1; : : : ;ng .3mC12kC1/.

(i) For anyD.I/–fusion disk forDI.L/, we have xL.I/ 1

m!2^{m}
X

J<D.I/

. 1/^{j}^{J}^{j}.logP0.DI.L/J//^{.}^{m}^{/} .modL.I//:

(ii) IfmDk.2/, then for anyD.I/–fusion disk forDI.L/, we have xL.I/D 1

k!2^{k}
X

J<D.I/

. 1/^{j}^{J}^{j}P_{0}^{.}^{k}^{/}.DI.L/J/2Z:

Theorem 1.3 follows directly from Theorems 1.1 and 1.2, since Milnor proved in [20,
Theorem 7] that xD_{I}.L/.D.I//D xL.I/ (note that L.I/DD_{I}.L/.D.I//, again
as a consequence of [20, Theorem 7]).

Theorem 1.3 implies the following corollary.

Corollary 1.4 All Milnor invariants of a link L vanish if and only if all linking numbers ofL are zero and P

J<D.I/. 1/^{j}^{J}^{j}P_{0}^{.}^{k}^{/}.DI.L/J/D0 for allk.2/ and
for all sequencesI with length kC1.

The rest of the paper is organized as follows. In Section 2, we review some elements of the theory of claspers, which is the main tool in proving our main results. In Section 3, we recall some properties of the HOMFLYPT polynomial of knots. In Section 4, we review Milnor invariants and string links and give a few lemmas. Section 5 is devoted to the proof of Theorem 1.1. In Section 6, we prove Theorem 1.2 and show, as a consequence, how to use the HOMFLYPT polynomial to distinguish string links up to link-homotopy. The paper is concluded by a simple example which illustrates the necessity of the assumptions required in our results.

Convention 1.5 In this paper, given a sequence I of elements of f1; : : : ;ng, the notationJ <I will be used for any subsequence J of I, possibly empty or equal toI itself. By J ŒI, we mean any subsequence J of I that is not I itself. We will use the notation fIgfor the subset of f1; : : : ;ngformed by all indices appearing in the sequence I, and jIjwill denote the length of the sequence I.

Acknowledgements The authors thank Kazuo Habiro for useful discussions. The first author is supported by the French ANR research project ANR-11-JS01-00201 and by the JSPS Invitation Fellowship for Research in Japan (Short Term) number S-10127.

The second author is partially supported by a JSPS Grant-in-Aid for Scientific Research (C) number 23540074.

This work is dedicated to Professor Shin’ichi Suzuki on his 70th birthday.

## 2 Some elements of clasper theory

The primary tool in the proofs of our results is the theory of claspers. We recall here the main definitions and properties of this theory that will be useful in subsequent sections.

For a general definition of claspers, we refer the reader to Habiro [9].

Definition 2.1 LetL be a (string) link. A surfaceG embedded in S^{3} (orD^{2}Œ0;1)
is called agraph clasperfor L if it satisfies the following three conditions:

(1) G is decomposed into disks and bands, callededges, each of which connects two distinct disks.

(2) The disks have either 1 or 3 incident edges, and are calledleavesorvertices respectively.

(3) G intersects L transversely, and the intersections are contained in the union of the interiors of the leaves.

In particular, if a graph clasper G is a disk, we call it atree clasper.

Throughout this paper, the drawing convention for claspers are those of [9, Figure 7], unless otherwise specified.

The degree of a connected graph clasper G is defined as half of the number of vertices and leaves. A tree clasper of degree k is called aCk–tree. Note that a Ck–tree has exactly .kC1/ leaves.

A graph clasper for a (string) link L issimpleif each of its leaves intersects L at exactly one point.

Let G be a simple graph clasper for an n–component (string) link L. Theindexof G is the collection of all integers i such that G intersects the i–th component of L. For example, if G intersects component 3 twice and components 2 and 5 once, and is disjoint from all other components of L, then its index isf2;3;5g.

Given a graph clasper G for a (string) link, there is a procedure to construct a framed
link, in a regular neighbourhood of G. There is thus a notion of surgery along G,
which is defined as surgery along the corresponding framed link. In particular, surgery
along a simple C_{k}–tree is a local move as illustrated in Figure 1.

The C_{k}–equivalence is the equivalence relation on (string) links generated by surg-
eries along connected graph claspers of degree k and isotopies. Alternatively, the
Ck–equivalence can be defined in term of “insertion” of elements of the k–th term of
the lower central series of the pure braid group; see Stanford [23]. We use the notation
LCk L^{0} for Ck–equivalent (string) linksL and L^{0}.

Figure 1. Surgery along a simpleC5–tree

It is known that the C_{k}–equivalence becomes finer as k increases, and that two links
areC_{k}–equivalent if and only if they are related by surgery along simpleC_{k}–trees [9].

Moreover, it was shown by Goussarov [6] and Habiro [9] that this equivalence relation
characterizes the topological information carried by finite type knot invariants. More
precisely, it is shown in [6; 9] that two knots are C_{k}–equivalent if and only if they
cannot be distinguished by any finite type invariant of degree <k.

2.1 Linear trees and planarity

For k3, a C_{k}–tree G having the shape of the tree clasper in Figure 1 is called a
linear C_{k}–tree. The left-most and right-most leaves of G in Figure 1 are called the
endsofG.

Now suppose that the G is a linearCk–tree for some knot K, and denote its ends by
f andf^{0}. Then the remaining .k 1/ leaves of G can be labelled from 1 to .k 1/,
by travelling along the boundary of the disk^{1} G from f to f^{0} so that all leaves are
visited. We say that G is planarif, when travelling along K from f to f^{0}, either
following or against the orientation, the labels of the leaves met successively are strictly
increasing. See Figure 2 for an example.

P N

Figure 2. TheC6–treeP is planar for the unknot, whileN is nonplanar.

2.2 Calculus of claspers for parallel claspers

We shall need refinements of [9, Propositions 4.4 and 4.6] for parallel tree claspers.

1Recall that a clasper is an embedded surface: in particular, sinceT is a tree clasper, the underlying surface is homeomorphic to a disk.

Here, byparalleltree claspers we mean a family ofmparallel copies of a tree clasperT, for somem1. We call mthemultiplicityof the parallel clasper. Note that there is no ambiguity in the notion of parallel copies here, since for a tree clasper the underlying surface is homeomorphic to a disk.

Lemma 2.2 Letm;k;k^{0}1 be integers. LetT be a parallelC_{k}–tree with multiplic-
ity mfor a (string) link L, and letT^{0} be aC_{k}0–tree forL, disjoint fromT.

(1) LetTz[ zT^{0}be obtained fromT [T^{0} by sliding a leaff^{0}of T^{0} overm parallel
leaves ofT (see Figure 3(a)). ThenLT[T^{0} is ambient isotopic toL_{T}_{z}_{[ z}_{T}_{0}_{[}_{Y}_{[}_{C},
whereY denotes the parallelC_{k}_{C}_{k}0–tree with multiplicitym obtained by insert-
ing a vertexv in the edgee of T and connecting v to the edge incident tof^{0}as
shown in Figure 3(a), and where C is a disjoint union ofCkCk^{0}C1–trees forL.
(2) LetTz[ zT^{0}be obtained fromT[T^{0}by passing an edge ofT^{0}acrossm parallel

edges ofT (see Figure 3(b)). Then LT[T^{0} is ambient isotopic toL_{T}_{z}_{[ z}_{T}_{0}_{[}_{H}_{[}_{C},
whereH denotes the parallel CkCk^{0}C1–tree with multiplicity m obtained by
inserting vertices in both edges, and connecting them by an edge as shown in
Figure 3(b), and whereC is a disjoint union ofC_{k}_{C}_{k}0C2–trees forL.

...

...

...

...

...

... ...

...

...

... ...

...

...

...

...

...

...

T^{0} T

e

f^{0}

m m m m m m

v

H

Tz^{0} Y Tz T^{0} T Tz^{0} Tz

(a) (b)

Figure 3. Leaf slide and crossing change involving parallel tree claspers

This result is well-known for mD1. The general case is easily proved using the arguments of the proof of [9, Propositions 4.4 and 4.6], respectively.

Remark 2.3 Notice that, following the proofs of [9, Propositions 4.4 and 4.6], the
index of each of the tree claspers involved in Lemma 2.2 can be determined from those
of T and T^{0} as follows. We have that the index ofTz is equal to the index of T, the
index of Tz^{0} is equal to the index of T^{0}, and the indices ofY, H and each connected
component of C are equal to the union of the indices ofT andT^{0}.

## 3 The HOMFLYPT polynomial

In this section, we recall the definition of the HOMFLYPT polynomial, and mention a few useful examples and properties.

TheHOMFLYPT polynomialP.LIt;z/2ZŒt^{˙}^{1};z^{˙}^{1}of an oriented linkL is defined
by the formulas

(1) P.UIt;z/D1,

(2) t ^{1}P.L_{C}It;z/ tP.L It;z/DzP.L0It;z/,

where U denotes the unknot and where L_{C}, L and L_{0} are three links that are
identical except in a 3–ball where they look as follows:

L_{C}D ; L D ; L0D :

In particular, the HOMFLYPT polynomial of an r–component link K is of the form P.KIt;z/D

N

X

kD1

P2k 1 r.KIt/z^{2k 1 r};

where P_{2k 1 r}.KIt/2 ZŒt^{˙}^{1} is called the .2k 1 r/–th coefficient polynomial
of K. Furthermore, the lowest degree coefficient polynomial of K is given by
(3-1) P1 r.KIt/Dt^{2}^{Lk.}^{L}^{/}.t ^{1} t/^{r 1}

r

Y

iD1

P0.KiIt/;

whereKi denotes the i–th component ofK, and where Lk.L/WDP

i<jlk.Li;Lj/; see [14, Proposition 22].

Denote by P_{k}^{.}^{l}^{/}.L/ the l–th derivative of P_{k}.LIt/ evaluated at t D1. It was proved
by Kanenobu and Miyazawa that P_{k}^{.}^{l}^{/} is a finite type invariant of degree kCl [12].

In particular, P_{0}^{.}^{l}^{/} is of degreel, and thus is an invariant of ClC1–equivalence.

It is well-known that the HOMFLYPT polynomial of knots is multiplicative under
connected sum. Thus the same holds for the lowest degree coefficient polynomial P0,
and in general, for any integer nand any two oriented knots K and K^{0}, we have

P_{0}^{.}^{n}^{/}.K]K^{0}/DP_{0}^{.}^{n}^{/}.K/CP_{0}^{.}^{n}^{/}.K^{0}/C

n 1

X

kD1

n k

P_{0}^{.}^{k}^{/}.K/P_{0}^{.}^{n k}^{/}.K^{0}/:

If, moreover, we assume that the knot K isCn–equivalent to the unknot, then we have
(3-2) P_{0}^{.}^{n}^{/}.K]K^{0}/DP_{0}^{.}^{n}^{/}.K/CP_{0}^{.}^{n}^{/}.K^{0}/;

since P_{0}^{.}^{k}^{/} is an invariant ofC_{k}_{C}_{1}–equivalence, for all k.

In general, a simple way to derive an additive knot invariant from the coefficient
polynomial P0 is to take its log. (SinceP0.KIt/ is in ZŒt^{˙}^{1}and P0.KI1/D1 for

any knot K, logP0.KIt/can be regarded as a smooth function defined on an open
interval which contains1.) Indeed, we have that, for any two oriented knotsK andK^{0},

.logP_{0}/.K]K^{0}It/D.logP_{0}/.KIt/C.logP_{0}/.K^{0}It/:

Denote by.logP0.K//^{.}^{n}^{/} the n–th derivative of logP0.KIt/ evaluated att D1. As
mentioned in the introduction, .logP0.K//^{.}^{n}^{/} is equal to P0.K/^{.}^{n}^{/} plus a sum of
products of P0.K/^{.}^{k}^{/}’s withk <n. So we see that .logP0/^{.}^{n}^{/} is an additive finite
type knot invariant of degree n, and thus is an invariant of CnC1–equivalence.

The following simple example shall be useful later.

Lemma 3.1 Let n1, and let "D."0; "1; : : : ; "n; "nC1/2 f 1;1g^{n}^{C}^{2}. Let K^{"}_{n} be
the knot represented in Figure 4. Then

.logP0.K_{n}^{"}//^{.}^{n}^{C}^{1}^{/}DP_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{"}/D. 1/^{n}2^{n}^{C}^{1}.nC1/!

nC1

Y

iD0

"i:

Notice that K^{"}_{n} isC_{n}_{C}_{1}–equivalent to the unknot, and that for allkn, we thus have
.logP_{0}.K_{n}^{"}//^{.}^{k}^{/}DP_{0}^{.}^{k}^{/}.K_{n}^{"}/D0.

Proof Let us prove the second equality. We first prove the formula for the knot
K_{n}^{C}WDK_{n}^{.}^{1}^{;:::;}^{1}^{/}, by induction onn. SinceK^{C}_{1} is the trefoil, we haveP_{0}^{.}^{2}^{/}.K_{1}^{C}/D 8.
Suppose that n>1. Clearly, changing the crossing c of K_{n}^{C} yields the unknot (see
Figure 4). Hence

P0.K^{C}_{n}It/Dt^{2}CtP 1.LnIt/;

whereLnDK1[K2 is the 2–component link represented on the right-hand side of
Figure 4. Notice that K1 is an unknot, while K2 is a copy of the knot K^{C}_{n 1}. Hence

=

;

= ...

...

...

...

...

...

K_{n}^{"}

K2

Ln K1

where C1

1 c

"0

"1 "2 "n 1 "n

"nC1

Figure 4. The knotK_{n}^{"} and the2–component linkLn

by Equation (3-1) we have

P0.K_{n}^{C}It/Dt^{2}C.1 t^{2}/P0.K_{n 1}^{C} It/:

By differentiating this equation .nC1/ times and evaluating att D1, we obtain
P_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/D 2.nC1/P_{0}^{.}^{n}^{/}.K_{n 1}^{C} / n.nC1/P_{0}^{.}^{n 1}^{/}.K^{C}_{n 1}/:

Since K^{C}_{n 1} isCn–equivalent to the unknot, we see that P_{0}^{.}^{n 1}^{/}.K_{n 1}^{C} /D0. Hence
we have

P_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/D 2.nC1/P_{0}^{.}^{n}^{/}.K_{n 1}^{C} /:

The induction hypothesis impliesP_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/D. 1/^{n}2^{n}^{C}^{1}.nC1/!.

Now, notice that in general K_{n}^{"} is obtained from the unknot by surgery along the
linearCnC1–tree represented in Figure 5. It follows from [9, Claim on page 36] that

...

=

= where C1

1

"0

"1 "2 "n

"nC1

Figure 5. Here, a on an edge represents a negative half-twist.

K_{n}^{"}]K_{n}^{C} is C_{n}_{C}_{2}–equivalent to the unknot (resp. to K_{n}^{C}]K_{n}^{C}) if QnC1

iD0 "^{i} is equal
to 1 (resp. to D1). SinceP_{0}^{.}^{n}^{C}^{1}^{/} is an invariant of C_{n}_{C}_{2}–equivalence and the knot
K_{n}^{"} is C_{n}_{C}_{1}–equivalent to the unknot for any"2 f 1;1g^{n}^{C}^{2}, we have

1C

nC1

Y

iD0

"i

P_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/DP_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{"}]K_{n}^{C}/DP_{0}^{.}^{n}^{C}^{1}^{/}.K^{"}_{n}/CP_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/

Hence we have

P_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{"}/DP_{0}^{.}^{n}^{C}^{1}^{/}.K_{n}^{C}/

nC1

Y

iD0

"i: The second equality follows.

Recall that .logP_{0}/^{.}^{n}^{C}^{1}^{/} is given by the sum of P_{0}^{.}^{n}^{C}^{1}^{/} and a combination of P_{0}^{.}^{k}^{/}’s
withkn. Since the knot K_{n}^{"} is C_{n}_{C}_{1}–equivalent to the unknot, the first equality
follows.

We note from the above proof that Lemma 3.1 gives the variation of .logP0/^{.}^{n}^{C}^{1}^{/} and
P_{0}^{.}^{n}^{C}^{1}^{/} under surgery along a planar linear CnC1–tree for the unknot. On the other
hand, the HOMFLYPT polynomial does not change under surgery along a nonplanar
tree clasper, as follows from a formula of Kanenobu [11].

Lemma 3.2 LetT be a nonplanar linear tree clasper for a knotK. ThenP0.KTIt/D P0.KIt/.

Proof We may assume thatKT andKare given by identical diagrams, except in a disk
where they differ as illustrated in Figure 6. LetLŒ"2; : : : ; "^{n} ."^{j}2f 1;1g;jD2; : : : ;n/

...

...

...

...

K

KT

cn1

cn2

c_{.}n 1/1

c_{.}n 1/2

c21

c22 c1

Figure 6. The knotsKandKT

be the link obtained from KT by smoothing the crossing c1, and
(i) smoothing the crossing c_{j 1} if"j D1, or

(ii) changing the crossing cj 1 and smoothing the crossingcj 2 if"j D 1.

Kanenobu showed that if all linksLŒ"2; : : : ; "^{n}have less thannC1components, then
P_{0}.K_{T}It/DP_{0}.KIt/; see [11, (3.9)]. Moreover, Kanenobu showed how to estimate
the number of components of LŒ"2; : : : ; "n using a kind of a chord diagram which
corresponds to the smoothed crossings. More precisely, the chord diagram associated
to LŒ"2; : : : ; "n represents the n–singular knot obtained from KT by changing the
crossing c1 and each crossingcj 1 (resp.cj 2) such that "j D1 (resp. "j D 1) into
double points. Kanenobu showed thatLŒ"2; : : : ; "n has nC1 components if and only
if the associated chord diagram contains no intersection among the chords; see the
proof of [11, Lemma (3.7)]. If T is nonplanar, then it is not hard to see that, for any
."j 2 f 1;1g; j D1;2; : : : ;n/, the corresponding chord diagram contains such an
intersection. Thus we have the conclusion.

Remark 3.3 Lemma 3.1 and Lemma 3.2 are related to the main results of Ka- nenobu [11] and Horiuchi [10].

## 4 Milnor invariants

4.1 A short definition

Given ann–component link L inS^{3}, denote by the fundamental group of S^{3}nL,
and by q theq–th subgroup of the lower central series of . We have a presentation
of=q with ngenerators, given by a choice of meridian mi of thei–th component
of L, i D1; : : : ;n. So the longitude j of the j–th component of L (1j n)
is expressed modulo q as a word in the mi’s (abusing notation, we still denote

this word by j). The Magnus expansion E.j/ of j is the formal power series
in noncommuting variables X_{1}; : : : ;Xn obtained by substituting 1CXi for mi and
1 XiCX_{i}^{2} X_{i}^{3}C for m_{i} ^{1}, 1i n.

Let IDi1i2: : :i_{k 1}j be a sequence of elements off1; : : : ;ng. Denote byL.I/the
coefficient of Xi1 Xik 1 in the Magnus expansion E.j/. TheMilnor invariant
xL.I/is the residue class ofL.I/modulo the greatest common divisor of all L.J/
such that J is obtained from I by removing at least one index and permuting the
remaining indices cyclicly [19; 20]. The indeterminacy comes from the choice of the
meridians mi. Equivalently, it comes from the indeterminacy of representing the link
as the closure of a string link [7]. Let us recall below the definition of these objects.

4.2 String links

Let n1, and letD^{2}R^{2}be the unit disk equipped with nmarked pointsx_{1}; : : : ;xn

in its interior, lying on the diameter on the x–axis of R^{2}. An n–string link, or n–

component string link, is the image of a proper embedding Fn

iD1Œ0;1i!D^{2}Œ0;1
of the disjoint union Fn

iD1Œ0;1i of n copies of Œ0;1 in D^{2}Œ0;1, such that for
each i, the image of Œ0;1i runs from .xi;0/ to .xi;1/. Each string of an n–string
link is equipped with an (upward) orientation. The n–string link fx1; : : : ;xng Œ0;1
in D^{2}Œ0;1 is called thetrivialn–string linkand is denoted by 1n.

For each marked pointxi2D^{2}, there is a pointyi on@D^{2}in the upper half ofR^{2}such
that the segment piDxiyi is vertical to the x–axis, as illustrated in Figure 8. Given
an n–string link LDSn

iD1Li in D^{2}Œ0;1R^{2}Œ0;1, theclosureLy of L is the
n–component link defined by LyDSn

iD1LyiDL[.Sn

iD1.pi f0;1g/[.yiŒ0;1//.
The set of isotopy classes ofn–string links fixing the endpoints has a monoid structure,
with composition given by thestacking productand with the trivial n–string link1n as
unit element. Given two n–string linksL and L^{0}, we denote their product byLL^{0},
which is obtained by stackingL^{0} above L and reparametrizing the ambient cylinder
D^{2}Œ0;1.

Habegger and Lin showed that Milnor invariants are actually well defined integer-valued invariants of string links [7]. (We refer the reader to [7] or Yasuhara [26] for a precise definition of Milnor invariants .I/ of string links.) Furthermore, Milnor invariants of length k are known to be finite-type invariants of degree k 1 for string links by Bar-Natan [1] and Lin [15]. As a consequence, Milnor invariants of length k for string links are invariants of Ck–equivalence. Habiro showed that the same actually holds for Milnor invariants of links [9].

4.3 Some results

It was shown by Habegger and Lin that Milnor invariants without repeated indices classify string links up to link-homotopy [7]. Here, the link-homotopy is the equivalence relation generated by self-crossing changes. In [27], the second author gave an explicit representative for the link-homotopy class of any n–string link in terms of linear tree claspers. We shall make use of this representative in this paper, and recall its definition below.

Let J_{k} denote the set of all sequences j_{0}j_{1}: : :j_{k} of kC1 nonrepeating integers
from f1; : : : ;ngsuch that j_{0}<jm<j_{k} for all m. Let i_{0}i_{1}: : :i_{k} be a sequence of
.kC1/ integers from f1; : : : ;ng such that i0<i1< <i_{k 1}<i_{k}, and let aJ be
a permutation of fi1; : : : ;i_{k 1}g. ThenJ Di0aJ.i1/ aJ.i_{k 1}/i_{k} is in J_{k} (and all
elements of J_{k} can be realized in this way). Let TJ be the simple linear C_{k}–tree
for1n as illustrated in Figure 7. Here, aJ is the unique positive k–braid which defines
the permutation aJ and such that every pair of strings crosses at most once. In the
figure, we also implicitly assume that all edges of TJ overpassall components of1n.
(This assumption is crucial in the computation of Milnor invariants.) Let TxJ be the

*...*

*...*

*...*

***

TJ

aJ

i1i2 i_{k 1}

i0 i_{k}

i1i2 i_{k 1}

Figure 7. TheC_{k}–treesTJ andTxJ

C_{k}–tree obtained fromTJ by inserting a positive half-twist in the–marked edge; see
Figure 7. Denote respectively byVJ and V_{J} ^{1} then–string links obtained from 1n by
surgery along TJ and TxJ. This notation is justified by the fact that, for anyJ in J_{k},
the string link VJ V_{J} ^{1} isCkC1–equivalent to the trivial one [9].

Theorem 4.1 [27] Anyn–string linkL is link-homotopic tolDl1 ln 1, where liD Y

J2J_{i}

V_{J}^{x}^{J};where xJ Dli.J/D

L.J/ ifiD1; L.J/ l1li 1.J/ ifi2:

Remark 4.2 The above statement slightly differs from the one in [27]. There, another family of Ck–trees is used in place of TJ and TxJ. However, the present statement is

shown by the exact same arguments as for [27, Theorem 4.3], since for anyJ;J^{0}2J_{k}
(k1), we have

(4-1) V_{J}.J^{0}/D

1 ifJ DJ^{0},
0 otherwise
(compare with [27, Lemma 4.1]).

For the representativelDl1 ln 1 above, we have the following lemmas.

Lemma 4.3 LetI<12: : :nwithjIj Dmn. Then,
(4-2) l.I/x_{I} .mod gcdfx_{J} jJ ŒIg/:

Moreover, for allk<m 1, we have

(4-3) l1lk.I/0.mod gcdfxJ jJ ŒI; jJj kC1g/:

Proof By [18, Lemma 3.3] and Theorem 4.1, we have l.I/ D l1lm 2.I/C
lm 1.I/Dl1lm 2.I/Cx_{I}. Hence (4-3) implies (4-2), and it suffices to prove
Equation (4-3).

Note that, for an n–string link L DSn

iD1Li, we have I.L/ DI.S

i2fIgLi/. Hence we may assume that ID12: : :m and thatl is an m–string link.

The result is shown by an analysis of the Magnus expansion of a longitude of each

“building block” V_{J}^{x}^{J}, for all k <m 1 and all J 2J_{k}. Since we are aiming at
computing Milnor invariant .I/, we compute up to termsO.I/involving monomials
Xi_{1}Xi_{2} Xip such that i_{1}i_{2}: : :ip is not a subsequence of 12: : : .m 1/. Note that
O.I/includes any monomial where some variable appears at least twice, as well as
any monomial involving Xm. For a subset fKgof fIg, we will also use the notation
M.fKg/ for a sum of terms involving monomials such that allXj .j 2 fKg/ appear
exactly once in each monomial.

Let J Dj_{0}j_{1}: : :j_{k} 2J_{k}, for some k<m 1. Let j be an index in I, and denote
by j the j–th longitude ofV_{J}^{˙}^{1}. Notice that all monomials appearing in the Magnus
expansion E.j/ are in the variables Xi such that i2 fJg, since all edges of both TJ

and TxJ overpass all components of 1n. There are three cases:

(i) Ifj <j_{0} orj_{k}<j, then clearly we haveE.^{j}/D1.

(ii) Ifj 2 fJg, since all Milnor invariants of V_{J}^{˙}^{1} with length at mostk vanish,
E.j/D1CM.fJg n fjg/CM.fJg/CO.I/:

(iii) If j … fJg and j0<j <jk, then since V_{J}^{˙}^{1}n.ith component/ is trivial for
anyi2 fJg, all (nontrivial) monomials appearing in E.^{j}/ contain all variables
Xi .i 2 fJg/. Hence we have a Magnus expansion of the form

E.j/D1CM.fJg/CO.I/:

Summarizing all three cases, we have (4-4) E.j/D

8

<

:

1 ifj<j0orj_{k}<j,
1CM.fJg n fjg/CM.fJg/CO.I/ ifj2 fJg,

1CM.fJg/CO.I/ ifj… fJgandj0<j<j_{k}.
Now let us consider the stacking product V_{J}^{˙}^{1}V_{J}^{˙}^{1}. The Magnus expansion of the
j–th longitude ofV_{J}^{˙}^{1}V_{J}^{˙}^{1}is given byE.j/E.zj/, wherej is thej–th longitude
of V_{J}^{˙}^{1} andE.zj/ is obtained fromE.j/ by replacing Xi with E.i/ ^{1}XiE.i/
for eachi 2 fIg. By (4-4), we have

E.i/ ^{1}XiE.i/D
8

<

:

Xi ifi<j0orjk <i, XiCM.fJg/CO.I/ ifi2 fJg,

XiCM.fJg [ fig/CO.I/ ifi… fJgandj0<i<jk.
This implies that the Magnus expansion of thej–th longitude ofV_{J}^{˙}^{1}V_{J}^{˙}^{1}is given by

E.j/E.zj/D 8

<

:

1 ifj <j_{0} orj_{k} <j,
E.j/E.j/CM.fJg/CO.I/ ifj 2 fJg;

1CM.fJg/CO.I/ ifj … fJgandj_{0}<j <j_{k}:
Generalizing this argument, we obtain that the Magnus expansion of the j–th longitude
J;j in V_{J}^{x}^{J} is given by

(4-5) E.J;j/D 8

<

:

1 ifj<j0orj_{k}<j,
E.j/^{j}^{x}^{J}^{j}CM.fJg/CO.I/ ifj2 fJg;

1CM.fJg/CO.I/ ifj… fJgandj0<j<j_{k}:
Let us now focus on the case j Dm. There are two cases to consider.

If m is not in fJg, then necessarily j_{k} < m and by (4-5) we have that the
Magnus expansion of them–th longitude J;m inV_{J}^{x}^{J} is1.

Ifmis in fJg, then necessarilyj_{k}Dm. Since j0 is the smallest integer infJg,
any monomial in M.fJg n fmg/ whose leftmost variable is not Xj0 belongs
toO.I/. Moreover, each term inM.fJg/involves the variable Xm, and hence
belongs to O.I/. So (4-1) and (4-4) show that the Magnus expansion of the
m–th longitude of V_{J}^{˙}^{1} isE.m/D1˙Xj0 Xjk 1CO.I/. Equation (4-5)
then implies that the Magnus expansion of the m–th longitude J;m inV_{J}^{x}^{J} is
given by1CxJXj0 Xjk 1CO.I/.

Summarizing, for jDm we obtain that E.J;m/D

8

<

:

1 ifm… fJg,

1CO.I/ .modx_{J}/ ifm2 fJgandJ <I,
1CO.I/ otherwise:

We can now complete the computation ofl1lk.I/. Since the Magnus expansion of the
m–th longitude of l1 l_{k} is obtained from a product of E.J;m/’s (fJg fIg; J 2
Sk

sD1J_{s}) by replacing each variableXi with XiC.monomials involvingXi/, it is
of the form

1CO.I/ .mod gcdfxJ jJ ŒI; jJj kC1g/:

This implies thatl1lk.I/0 .mod gcdfxJ jJ ŒI; jJj kC1g/, as desired.

Lemma 4.4 LetI<12: : :nwithjIj Dmn. Then

l.I/Dgcdfx_{J} jJ ŒIg:

Proof The proof is by induction on m. For mD3, the result is clear since xij D l1.ij/D l.ij/ for any i;j. Now, let m 4. It will be convenient to use the notation ık.I/ for the set of all sequences of length .m k/ obtained from I by removing k indices and permuting cyclicly. By definition,

l.I/Dgcd.fl.J/jJ 2ık.I/; k>1g [ fl.J/jJ 2ı1.I/g/:

By the induction hypothesis, we have that

gcdfl.J/jJ 2ık.I/; k>1g Dgcdfl.J/jJ 2ı1.I/g

Dgcdfl.J/jJ 2ı1.I/; J <Ig
DgcdfxJ^{0}jJ^{0}ŒJ; J 2ı1.I/; J <Ig
DgcdfxJ^{0}jJ^{0}<I; jJ^{0}j<m 1g:

On the other hand, by Lemma 4.3, for allJ2ı1.I/ .J<I/and for any sequence.J/ obtained from J by permuting cyclicly, we have

l..J//l.J/xJ .mod gcdfxJ^{0}jJ^{0}ŒJg.Dl.J///:

It follows that l.I/Dgcdfx_{J} jJ <I; jJj m 1g, as desired.

## 5 Proof of Theorem 1.1

Let LDSn

iD1Li be ann–component link inS^{3}, and letI be a sequence of.mC1/
distinct elements of f1; : : : ;ng. It is sufficient to consider here the case mC1Dn,

since, if mC1<n, we have that xL.I/D x^{S}_{i}_{2f}_{I}_{g}Li.I/. We may further assume
that ID12: : :nwithout loss of generality. Indeed, for any permutationI^{0}of 12: : :n,
we have thatxL.I^{0}/D xL^{0}.12: : :n/, whereL^{0} is obtained from L by reordering the
components appropriately.

We first show how to reformulate the problem in terms of string links.

5.1 Closing string links into knots

Let B_{I} be anI–fusion disk forL, as defined in the introduction. Up to isotopy, we
may assume that the 2n–gon BI lies in the unit disk D^{2} as shown in Figure 8, where
the edges pj are defined by pj Dxjyj,1j n. We may furthermore assume that
L[BI lies in the cylinder D^{2}Œ0;1, such thatBI .D^{2} f0g/, and such that

L\@.D^{2}Œ0;1/D

n

[

jD1

.pj f0g/[.fyjg Œ0;1/[.pj f1g/

:

00 0 11

01 100

1100 11

00 11

0000 1111 00

11 00 11

BI

x1x2x3 xn

y1

y2y3

yn

Figure 8. The2n–gonBI lying in the unit diskD^{2}

In this way, we obtain an n–string link whose closure y is the link L, by setting
(5-1) WDLn.L\@.D^{2}Œ0;1//:

For example, Figure 9 represents a 3–string linkˇ whose closure is the Borromean rings (there, the dotted part represents the intersection of the Borromean rings and

@.D^{2}Œ0;1/).

Given an n–string link KDSn

iD1Ki and any subsequence J of ID12: : :n, we will denote by K.J/ the knot

K.J/WD

[

j2fJg

Kyj

[@B_{I}

n

[

j2fJg

Kyj

\B_{I}

:

ˇD

BID^{2} f0g

Figure 9. The string linkˇ, whose closure is the Borromean rings

For example, Figure 10 represents the knots ˇ.13/ and ˇ.123/ obtained from the 3–string link ˇ shown in Figure 9.

ˇ.13/D D ˇ.123/D D

Figure 10. The knots ˇ.13/andˇ.123/for the 3–string linkˇ

Note thatK.J/coincides with the knot.Ky/J defined in the introduction for the choice of I–fusion disk BI specified above.

Recall from Section 4.3 that, for any k and any J 2J_{k}, VJ (resp. V_{J} ^{1}) denotes
the n–string link obtained from1n by surgery along the C_{k}–treeT_{J} (resp. Tx_{J}); see
Figure 7. Denote by tJ (resp. xtJ) the image of the C_{k}–tree TJ (resp. TxJ) for 1n

under taking the closure V_{J}^{˙}^{1}.I/. We observe thattJ (resp.xtJ) is a planar tree clasper
for the unknot if and only if J <I. In this case, note that VJ.I/ (resp. V_{J} ^{1}.I/) is
the knotK_{k 1}^{"} of Figure 4 with "D. ;C; : : : ;C/ (resp. for "D.C;C; : : : ;C/). In
particular, observe that V_{I}^{x}^{I}.J/is the unknot for all J ŒI and that, by Lemma 2.2
(formD1), the knot V_{I}^{x}^{I}.I/ is Cn–equivalent to the connected sum ofjxIj copies
of V_{I}^{"}^{I}.I/, where "I denotes the sign of xI. By Lemmas 3.1 and 3.2 we thus have,
for all J 2J_{n 1},

(5-2)

.logP_{0}.V_{J}^{x}^{J}.I///^{.}^{n 1}^{/}DP_{0}^{.}^{n 1}^{/}.V_{J}^{x}^{J}.I//

D

. 1/^{n 1}xI.n 1/!2^{n 1} ifJ DI;

0 otherwise.

5.2 Proof of Theorem 1.1

Let be the n–string link with closure L defined in Section 5.1. By Theorem 4.1,
is link-homotopic to l_{k} ln 1, where li DQ

J2J_{i}V_{J}^{x}^{J} is defined in Section 4.3,
and with n2kC1 (by our vanishing assumption on Milnor invariants). Hence
is obtained from lk ln 1 by surgery along a disjoint union R1 of simple C1–trees
whose leaves intersect a single component oflk ln 1.

By Lemma 2.2, for all J <I, we have that

.J/Cnln 1.J/ ] .lk ln 2/R1.J/:

Since .logP_{0}/^{.}^{n 1}^{/} is an invariant of C_{n}–equivalence for all n, it follows from the
additivity property of .logP_{0}/ that

logP_{0}..J//_{.}n 1/

D logP_{0}.l_{n 1}.J//_{.}n 1/

C logP_{0} .l_{k} l_{n 2}/R1.J/ ^{.}^{n 1}^{/}:
The proof of the next lemma is postponed to Section 5.3.

Lemma 5.1
1
.n 1/!2^{n 1}

X

J<I

. 1/^{j}^{J}^{j} logP0 .l_{k} ln 2/R_{1}.J/ ^{.}^{n 1}^{/}0 .modL.I//:

It follows from Equation (5-2) that 1

.n 1/!2^{n 1}
X

J<I

. 1/^{j}^{J}^{j} logP_{0}..J//_{.}n 1/

1

.n 1/!2^{n 1}
X

J<I

. 1/^{j}^{J}^{j} logP_{0}.l_{n 1}.J//_{.}n 1/ .modL.I//

. 1/^{n 1}

.n 1/!2^{n 1} logP0.V_{I}^{x}^{I}.I//_{.}n 1/ .modL.I//

x_{I} .modL.I//:

On the other hand, by Lemma 4.3 and Lemma 4.4, we have

L.I/D_{}.I/Dlkln 1.I/x_{I} .modL.I//;

which completes the proof.