ON BRAIDS AND LINKS UP TO LINK-HOMOTOPY EMMANUEL GRAFF Abstract. This paper deals with links and braids up to link-homotopy studied from the viewpoint

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ON BRAIDS AND LINKS UP TO LINK-HOMOTOPY

EMMANUEL GRAFF

Abstract. This paper deals with links and braids up to link-homotopy, studied from the viewpoint

of Habiro’s clasper calculus. More precisely, we use clasper homotopy calculus in two main directions.

First, we deﬁne and compute a faithful linear representation of the homotopy braid group, by using

claspers as geometric commutators. Second, we give a geometric proof of Levine’s classiﬁcation of

4-component links up to link-homotopy, and go further with the classiﬁcation of 5-component links in

the algebraically split case.

1. Introduction

The notion of link-homotopy was introduced in 1954 by J.W. Milnor in [22], in the context of knot

theory. It is an equivalence relation on links that allows continuous deformations during which two

distinct components remain disjoint at all times, but each component can self-intersect. Any knot

is link-homotopic to the trivial one, but for links with more than one component this equivalence

relation turns out to be quite rich and intricate. Since Milnor’s seminal work, link-homotopy has

been the subject of numerous works in knot theory see e.g. [6, 17, 24, 8], but also more generally in

the study of codimension 2 embeddings (and in particular knotted surfaces in dimension 4) [20, 3, 2]

and link-maps (self-immersed spheres) [4, 14, 15, 25]. In this paper we are interested in the study of

link-homotopy for braids and links.

The homotopy braid group has been studied by many authors. In [6] Goldsmith gives an example

of a non-trivial braid up to isotopy that is trivial up to link-homotopy; she also gives a presentation of

the homotopy braid group. A representation of the homotopy braid group is given by Humphries in

[13]. He uses it to show that the homotopy braid group is torsion-free for less than 6 strands. Finally

the pure homotopy braid group has been studied by Habegger and Lin in [8] as an intermediate object

for the classiﬁcation of links up to link-homotopy. As further developed below, our ﬁrst main result

is another linear representation of the homotopy braid group (Theorem 3.23), which we prove to be

faithful (Theorem 3.31) and which is computed explicitly in Theorem 3.26.

We also address the problem initially posed by Milnor in [22], of classifying links in the 3-sphere up

to link-homotopy. Milnor himself answered the question for the 2 and 3-component case. Furthermore,

Habegger and Lin [8] proposed a complete classiﬁcation, using a subtle algebraic equivalence relation

on pure braids, where two equivalent braids correspond to link-homotopic links. A more direct

algebraic approach had been proposed by Levine [17] just before the work of Habegger–Lin in the

4-component case. Our second main result is a new geometric proof of Levine’s classiﬁcation of

4-component links up to link-homotopy (Theorem 4.7). This approach seems to apply, at least in

principle, to links with a higher number of components: we illustrate this in Theorem 4.10 with

Date: March 1, 2023.

2020 Mathematics Subject Classiﬁcation. 57K10 20F36,

Key words and phrases. Links, Braid groups, Link-homotopy, Claspers.

arXiv:2210.01539v2 [math.GT] 28 Feb 2023

2 EMMANUEL GRAFF

the case of algebraically split 5-component links (that is, 5-component links with vanishing linking

numbers).

The notion of clasper was developed by Habiro in [9]. These are surfaces in 3–manifolds with some

additional structure, on which surgery operations can be performed. In [9], Habiro describes the

clasper calculus up to isotopy, which is a set of geometric operations on claspers that yield equivalent

surgery results. It is well known to experts how clasper calculus can be reﬁned for the study of

knotted objects up to link-homotopy (see for example [5, 26]). This homotopy clasper calculus, which

we review in Section 2, will be the key tool for proving all the main results outlined above.

The rest of this paper consists of three sections.

In Section 2, we review the homotopy clasper calculus: after brieﬂy recalling from [9] Habiro’s

clasper theory, we recall how a fundamental lemma from [5], combined with Habiro’s work, produces

a set of geometric operations on claspers having link-homotopic surgery results.

Section 3 is dedicated to the study of braids up to link-homotopy. We start by reinterpreting braids

in terms of claspers. In Section 3.1 we deﬁne comb-claspers, a family of claspers corresponding to

braid commutators. They are next used to deﬁne a normal form on homotopy braids, thus allowing

us to rewrite any braid as an ordered product of comb-claspers. In Section 3.2 after a short algebraic

interlude, we give a presentation of the pure homotopy braid group (Corollary 3.20), using the work

of [6] and [23] as well as the technology of claspers. Finally, we deﬁne and study in Section 3.3 a

representation of the homotopy braid group which is in a sense the linearization of the homotopy

Artin representation. We give its explicit computation in Theorem 3.26 (see also Example 3.28 for

the 3-strand case) and show its injectivity in Theorem 3.31. Moreover, from the injectivity of the

representation follows the uniqueness of the normal form and thus the deﬁnition of the clasp-numbers,

a collection of braid invariant up to link-homotopy. Note that our representation has lower dimension

than Humphries one. The correspondence between the two representations has not been established

yet, but we wonder if our representation could open new leads on the torsion problem for more than

six strands.

The ﬁnal Section 4 focuses on the study of links up to link-homotopy. The method used is based

on the precise description of some operations, which generate the algebraic equivalence relation men-

tioned above in the classiﬁcation result of Habegger and Lin [8]; we provide them with a topological

description in terms of claspers. This new point of view allows us, for a small number of components,

to describe when two braids in normal form have link-homotopic closures. We translate in terms of

clasp-number variations the action of those operations on the normal form. In this way we recover the

classiﬁcation results of Milnor [22] and Levine [17] for 4 or less components (Theorem 4.7). Moreover,

we also classify 5-component algebraically split links up to link-homotopy (Theorem 4.10).

Acknowledgement: The author thanks the referee for his/her careful reading and insightful sugges-

tions. This work is partially supported by the project AlMaRe (ANR- 19-CE40-0001-01) of the ANR.

The author thanks P. Bellingeri and J.B. Meilhan for their great advises and helpful discussions.

2. Clasper calculus up to link-homotopy

Clasper calculus has been developed by Habiro in [9] in the context of tangles up to isotopy.

Claspers turn out to be in fact a powerful tool to deal with link-homotopy. In this section we ﬁrst

deﬁne claspers and their associated vocabulary. Then we describe how to handle claspers up to

link-homotopy.

ON BRAIDS AND LINKS UP TO LINK-HOMOTOPY 3

2.1. General deﬁnitions. Let Mbe a smooth compact and oriented 3-manifold.

Deﬁnition 2.1. An n-component tangle in Mis a smooth embedding of an n-component ordered

and oriented 1-manifold (a disjoint union of circles and intervals) into M.

‚We say that two tangles are isotopic if they are related by an ambient isotopy of Mthat ﬁxes

the boundary.

‚We say that two tangles are link-homotopic if there is a homotopy between them ﬁxing the

boundary, and such that the distinct components remain disjoint during the deformation.

Deﬁnition 2.2. A disk Tsmoothly embedded in Mis called a clasper for a tangle θif it satisﬁes the

following three conditions:

-Tis the embedding of a connected thickened uni-trivalent graph with a cyclic order at each

trivalent vertex. Thickened univalent vertices are called leaves, and thickened trivalent ver-

tices, nodes.

-θintersects Ttransversely, and the intersection points are in the interior of the leaves of T.

- Each leaf intersects θin at least one point.

Diagrammatically a clasper is represented by a uni-trivalent graph corresponding to the one to be

thickened. The trivalent vertices are thickened according to Figure 1. On the univalent vertices we

specify how the corresponding leaves intersect θ, and we also indicate how the edges are twisted using

markers called twists (see Figure 1).

Figure 1. Local diagrammatic representation of claspers.

Deﬁnition 2.3. Let Tbe a clasper for a tangle θ. We deﬁne the degree of Tdenoted degpTqas its

number of nodes plus one, or equivalently its number of leaves minus one. The support of Tdenoted

supppTqis deﬁned to be the set of the components of θthat intersect T.

Deﬁnition 2.4. A clasper Tfor a tangle θis said to be simple if every leaf of Tintersects θexactly

once. A leaf of a simple clasper intersecting the l-th component is called an l-leaf.

Deﬁnition 2.5. We say that a simple clasper Tfor a tangle θhas repeats if it intersects a component

of θin at least two points.

Given a disjoint union of claspers Ffor a tangle θ, there is a procedure called surgery detailed in

[9] to construct a new tangle denoted θF. We illustrate on the left-hand side of Figure 2 the eﬀect of

a surgery on a clasper of degree one. Now if Fcontains some claspers with degree higher than one,

we ﬁrst apply the rule shown on the right-hand side of Figure 2, at each trivalent vertex: this breaks

up Finto a disjoint union of degree one claspers, on which we can perform surgery.

Note that clasper surgery commutes with ambient isotopy. More precisely for ian ambient isotopy

and Fa disjoint union of claspers for a tangle θwe have that ipθFq“pipθqqipFq. This is an elementary

example of clasper calculus, which refers to the set of operations on unions of a tangles with some

claspers, that allow to deform one into another with isotopic surgery result. These operations are

developed in [9], and we give in the next section the analogous calculus up to link-homotopy.

4 EMMANUEL GRAFF

Figure 2. Rules of clasper surgery.

2.2. Clasper calculus up to link-homotopy. In the whole section, Tand Sdenote simple claspers

for a given tangle θ. We use the notation T„S, and say that Tand Sare link-homotopic when the

surgery results θTand θSare so. For example if iis an ambient isotopy that ﬁxes θ, then T„ipTq.

Moreover, if θTis link-homotopic to θ, we say that Tvanishes up to link-homotopy and we denote

T„ H.

We begin by recalling a fundamental lemma from [5]; more precisely, the next result is the case

k“1 of [5, Lemma 1.2], where self C1-equivalence corresponds to link-homotopy.

Lemma 2.6. [5, Lemma 1.2] If Thas repeats then Tvanishes up to link-homotopy.

It is well known to the experts that combining Lemma 2.6 with the proofs of Habiro’s technical

results on clasper calculus [9], yields the following link-homotopy clasper calculus.1

Proposition 2.7. [9, Proposition 3.23, 4.4, 4.5 and 4.6] We have the following link-homotopy equiv-

alences (illustrated in Figure 3).

(1) If Sis a parallel copy of Twhich diﬀers from Tonly by one twist (positive or negative), then

SYT„ H.

(2) If Tand Shave two adjacent leaves and if T1YS1is obtained from TYSby exchanging these

leaves as depicted in (2) from Figure 3, then TYS„T1YS1Y˜

T, where ˜

Tis as shown in the

ﬁgure.

(3) If T1is obtained from Tby a crossing change with a strand of the tangle θas depicted in (3)

from Figure 3, then T„T1Y˜

T, where ˜

Tis as shown in the ﬁgure.

(4) If T1YS1is obtained from TYSby a crossing change between one edge of Tand one of Sas

depicted in (4) from Figure 3, then TYS„T1YS1Y˜

T, where ˜

Tis as shown in the ﬁgure.

(5) If T1is obtained from Tby a crossing change between two edges of Tthen T„T1.

❑∼❑

θθ

T ’T ~

❑∼❑

T ’

S’

(1)(1) (2)(2) (3)(3)

❑∼❑

(4)(4) (5)(5)

S’T ’ST T ’T ’TT

θθ

Figure 3. Basic clasper moves up to link-homotopy.

1Those moves are contained in [26] and [21] together with [5].

ON BRAIDS AND LINKS UP TO LINK-HOMOTOPY 5

Idea of proof. The result of [9] used here are up to Ck-equivalence, that is, up to claspers of degree up

to k. The key observation is that, by construction, all such higher degree claspers have same support

as the initial ones, hence they are claspers with repeats. Lemma 2.6 then allows to delete them up to

link-homotopy. 

Remark 2.8. Lemma 2.6 combined with Proposition 2.7 gives us some further results:

- First, statement p4qimplies that if |supppTq X supppSq| ě 1then we can realize crossing

changes between the edges of Tand S.

- Moreover, if |supppTq X supppSq| ě 2thanks to statement (2) we can also exchange the leaves

of Tand S.

- Furthermore, statement (3) allows crossing changes between Tand a component of θin the

support of T

Indeed, in each case the clasper ˜

Tinvolved in the corresponding statement has repeats and can thus

be deleted up to link-homotopy.

The next remark describes how to handle twists up to link-homotopy.

Remark 2.9. We have the following link-homotopy equivalences (illustrated in Figure 4).

(6) If T1is obtained from Tby moving a twist across a node then T„T1.

(7) If Tand T1are identical outside a neighborhood of a node, and if in this neighborhood Tand

T1are as depicted in (8) from Figure 4, then T„T1.

❑∼❑

(7)

❑∼❑

(6)

T ’T ’T ’T ’T ’T ’T ’T ’T ’T ’T T T T T T T T T T T ’T ’T ’T ’T ’T ’T ’T ’T ’T ’T T T T T T T T T T

Figure 4. How to deal with twist up to link-homotopy.

Remark 2.10. Remark 2.9 allows us to bring all the twists on a same edge and then cancel them

pairwise. Therefore we can consider only claspers with one or no twist.

Proposition 2.7 together with Remark 2.9 give us most of the necessary tools to understand clasper

calculus up to link-homotopy. The missing ingredient is the relation IHX which we give in the following

proposition.

Proposition 2.11. [9] Let TI,TH,TXbe three parallel copies of a given simple clasper that coincide

everywhere outside a 3-ball, where they are as shown in Figure 5. Then TIYTHYTX„ H. We say

that TI,THand TXverify the IHX relation.

TITH

THTX

Figure 5. The IHX relation for claspers.

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ONBRAIDSANDLINKSUPTOLINK-HOMOTOPYEMMANUELGRAFFAbstract.Thispaperdealswithlinksandbraidsuptolink-homotopy,studiedfromtheviewpointofHabiro'sclaspercalculus.Moreprecisely,weuseclasperhomotopycalculusintwomaindirections.First,wedeneandcomputeafaithfullinearrepresentationofthehomotopybraidgroup,byusingc...

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