THE BURAU REPRESENTATION AND SHAPES OF POLYHEDRA ETHAN DLUGIE Abstract. We use a geometric approach to show that the reduced Burau representation special-

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THE BURAU REPRESENTATION AND SHAPES OF POLYHEDRA

ETHAN DLUGIE

Abstract. We use a geometric approach to show that the reduced Burau representation special-

ized at roots of unity has another incarnation as the monodromy representation of a moduli space

of Euclidean cone metrics on the sphere, as described by Thurston. Using the theory of orbifolds,

we leverage this connection to identify the kernels of these specializations in some cases, partially

addressing a conjecture of Squier. The 4-strand case is the last case where the faithfulness ques-

tion for the Burau representation is unknown, a question that is related e.g. to the question of

whether the Jones polynomial detects the unknot. Our results allow us to place the kernel of this

representation in the intersection of several topologically natural subgroups of B4.

1. Introduction

In this paper, we consider two representations of groups arising in low dimensional topology. First

is the (reduced) Burau representation of braid groups

βn∶Bn→GLn−1(Z[t±])

that has been studied for almost a century [Bur35]. Second is a monodromy representation of

punctured sphere mapping class groups coming from a geometric structure on the moduli space of

Euclidean cone spheres,

ρ⃗

k∶Mod(S0,m;⃗

k)→PU(1, m −3),

as described by Thurston in [Thu98]. It has been found using algebraic techniques that these

seemingly disparate representations are quite closely related in that the latter is, in a sense, a

specialization of the former [McM13, Ven14]. Our ﬁrst theorem is a slight rephrasing of those

results, which we will establish in this work via geometric means. See the beginning of Section 2

for an introduction to the terminology of Euclidean cone metrics used in the following statement.

Theorem 1.1 (The Burau representation and polyhedra monodromy).Fix a choice of curvatures

⃗

k, which is to say a tuple of real numbers ⃗

k=(k1,...,km)with each 0<ki<2πand ∑m

i=1ki=4π.

Suppose further that nof these curvatures are equal, say k1=⋯=knwith n≤m−1, and write

k∗∈(0,2π)for this common value. Set q=exp(i(π−k∗)). Then the following diagram commutes

Bnβn(Bn)GLn−1(Z[t±])

Mod(S0,m;⃗

k)PU(1, m −3)PGLm−2(C)

ι∗

βn

ev(−q)

⊂

ρ⃗

k⊂

where βnis the n-strand (reduced) Burau representation, ρ⃗

kis the monodromy representation of

the moduli space of cone metrics, ι∗is the map on mapping class groups induced by an inclusion of

an n-times marked disk ι∶Dn↪S0,m, and ev(−q)is a slight alteration of an evaluation map to be

deﬁned in Deﬁnition 3.3.

arXiv:2210.06561v2 [math.GT] 9 Jul 2023

2 ETHAN DLUGIE

In the case where m=n+1 and Dnis included into an (n+1)-times punctured sphere, the

evaluation map mentioned in this theorem really is just an evaluation. This allows us to realize the

“specialized” Burau representation β(−q), where the formal variable tis evaluated at a given unit

complex number −q, as one of these polyhedral monodromy representations.

Corollary 1.2. Let ⃗

k=(k1,...,kn+1)be as in Theorem 1.1 with k∗=k1=⋯=kn. Write q=

exp(i(π−k∗)) Then the following diagram commutes

BnGLn−1(C)

Mod(S0,n+1;⃗

k)PU(1, n −2)PGLn−1(C)

ι∗

β(−q)

ρ⃗

k⊂

where Mod(S0,n+1;⃗

k)is the subgroup of the mapping class group of the (n+1)-times punctured

sphere that preserves the (n+1)-st point and may freely permute the other points.

This yields a containment ker(β(−q))≤nclBn(˜

S)⋅⟨τn⟩where ˜

Sis a lift of a normal generatoring

set for ker(ρ⃗

k)and τn∈Bnis the full twist braid on n-strands that generates the center of the braid

group.

In the statement, and in the rest of the paper, the notation nclG(S)indicates the normal closure

of a set Sinside of a group G. We will also write τp∈Bnfor a full twist about a curve surrounding

ppoints in the n-punctured disk. Any two such twists are conjugate in the braid group.

In his inﬂuential paper [Squ84], Squier brieﬂy considered the specializations of the Burau represen-

tation at roots of unity. He made a conjecture about the form that the kernels of such specializations

would take. We cannot verify Squier’s conjecture in the form that he stated it,1but using Corol-

lary 1.2, we are able to identify the kernel of these specializations in several cases.

Theorem 1.3 (Burau at roots of unity).Let qbe a primitive d-th root of unity and let β(−q)∶

Bn→GLn−1(C)denote the specialization of the Burau representation at t=−q. Then we have

(1) ker(β(−q))=nclBn(σd, τj

n−1)⋅⟨τℓ

n⟩

for the following values of n, d, j, l:

n4 5 6 7 8 9 10

d5 6 7 8 9 10 12 18 4 5 6 8 4 5 3 4 3 3 3

j∞ ∞ 14 8 6 5 4 3 ∞53242∞2 6 3 2

l5 3 7 4 9 5 3 9 4 2 3 8 2 5 6 4 3 2 3

Here σ∈Bndenotes one of the half-twist generators of Bn(all of which are conjugate), τn−1∈Bn

denotes a full twist on a curve surrounding n−1points in the punctured disk (all of which are

conjugate), and τn∈Bndenotes the full twist on the boundary of the punctured disk (which generates

the center of Bn). In a case with j=∞, we mean that the kernel is ncl(σd)⋅⟨τℓ

n⟩with no power of

τn−1.

We can also use the same method to identify the kernel of β(−q)in all cases with n=3 and d≥7.

The result is given in Theorem 5.4 and corrects the statement of [FK14, Theorem 1.2].

Whether or not the Burau representation is faithful is a natural question to ask. At present, the

answer is unknown only in the n=4 case, and this question has direct connections to the question

of whether the Jones polynomial detects the unknot [Big02, Ito15]. An element of the kernel of β4

1See Section 6 for a discussion on this point.

THE BURAU REPRESENTATION AND SHAPES OF POLYHEDRA 3

must also lie in the kernel of every specialization. Thus Theorem 1.3 as a direct corollary restricts

the kernel of β4to live in the intersection of several topologically natural normal subgroups of the

braid group. One should note however that the intersection of these ﬁnitely many subgroups is still

nontrivial by [Lon86, Lemma 2.1], so this alone is not enough to establish faithfulness.

Corollary 1.4 (Narrowing ker(β4)).Let β4∶B4→GL3(Z[t±]) denote the reduced Burau repre-

sentation of the 4-strand braid group. Then

ker(β4)≤nclB4(σd, τ ℓ

3)⋅⟨τℓ

4⟩

for powers d, j, ℓ as indicated in the table in Theorem 1.3. All eight of these normal subgroups have

inﬁnite index in B4.

In fact all of the normal subgroups of braid groups given by Theorem 1.3 have inﬁnite index in their

respective braid groups. I comment on the relationship between this and some remarkable work of

Coxeter [Cox59] in Section 6.

Some history and context. The question of the faithfulness of the Burau representation has

persisted since the representation was ﬁrst deﬁned nearly a century ago [Bur35]. Faithfulness is

easily shown for n=2,3 (see e.g. [Bir75, Theorem 3.15]). Faithfulness for other cases remained

open for several decades. Squier put forth two conjectures [Squ84, (C1) and (C2)] that, if both

true, would yield the faithfulness of the Burau representation. However, Moody found the Burau

representation to be nonfaithful for n≥10 [Moo93], and this result was quickly lowered to n≥6 by

Long and Paton [LP93]. A few years later, Bigelow found a simpler example of an element in the

kernel of β6and furthermore found that the Burau representation is not faithful for n=5 [Big99].

Funar and Kohno proved Squier’s conjecture (C2) in [FK14], so we know for all n≥5 that Squier’s

conjecture (C1) is false for almost all (even) values of d. At the time of writing of this article, the

faithfulness question is only open in the n=4 case.

Braid groups are already known to be linear by another representation, the Lawrence-Krammer

representation. See [Kra02] for an algebraic treatment of this result and [Big00] for a topological

proof. Yet the faithfulness of the Burau representation, especially in the n=4 case, is still of interest

due to its connection with the Jones polynomial in knot theory. Nonfaithfulness of β4implies that

the Jones polynomial fails to detect the unknot [Big02, Ito15]. There has been work on the n=4

question in the last few decades. For instance, a computer search by Fullarton and Shadrach shows

that a nontrivial element in the kernel of β4would have to be exceedingly complicated [FS19],

suggesting faithfulness. On the other hand, Cooper and Long found that β4is not faithful when

taken with coeﬃcients mod 2 and with coeﬃcients mod 3 [CL97, CL98].

Thurston’s work in [Thu98] was a geometric reframing of the monodromy of hypergeometric func-

tions considered by Deligne and Mostow in [DM86]. The algebro-geometric approach to studying

these monodromy representations has continued, notably in works such as [McM13] and [Ven14].

The analysis of Euclidean cone metrics on surfaces was extended by Veech [Vee93] and is still today

an active area of research in low-dimensional topology and dynamical systems.

Organization of the paper. The rest of the paper is organized as follows.

●Section 2 introduces Euclidean cone metrics on the sphere, and we construct explicit com-

plex projective coordinates on the moduli space.

●In Section 3, we prove Theorem 1.1 and Corollary 1.2 that allow us to relate the Burau

representation at roots of unity with the monodromy representation of the moduli spaces

of Euclidean cone metrics. Our proof uses the complex projective coordinates deﬁned in

Section 2.

4 ETHAN DLUGIE

●In Section 4 we gather several results about the complex hyperbolic geometry of the moduli

space and facts about geometric orbifolds.

●In Section 5 we prove Theorem 1.3 identifying the kernel of the Burau representation at

some roots of unity. This uses Corollary 1.2 with the results of Section 4. We also present

the application of these ideas to the β3case in Section 5.2.

●Section 6 contains a discussion of limitations of this work and several possible future direc-

tions and connections that I hope can spark further research with these techniques.

Acknowledgements. Substantial thanks to my advisor, Ian Agol, who ﬁrst informed me about the

connection between the Burau representation and the polyhedra monodromy of Thurston’s work.

Special thanks as well to both Nancy Scherich and Sam Freedman for repeated helpful conversations

through early drafts about the framing and phrasing of these results. Thanks to many others for

taking the time to read and comment on a preprint of this work. Thanks to Louis Funar and

Toshitake Kohno for graciously accepting a correction to one of their previous statements. And

ﬁnally, I thank the anonymous referee for a speedy review of and many inﬂuential comments on my

initial submission. In particular, the referee taught me about the conjectures of Squier which have

come to be a focal point of this work in its current form.

The author’s work was supported in part by a grant from the Simons Foundation (Ian Agol,

#376200).

2. Euclidean Cone Metrics on S2

Here we recall the moduli space of Euclidean cone metrics on the sphere. We describe local coordi-

nates on the moduli space into complex projective space. The construction is used in the proof of

Theorem 1.1 in Section 3.

Following [Thu98], we consider Euclidean cone metrics on the sphere. Such a metric is ﬂat every-

where on the sphere away from some number of singular cone points b1,...,bm. Around each cone

point bione sees some cone angle not equal to the usual 2πthat one ﬁnds around a smooth point.

Deﬁne the curvature kiat bito be the angular defect of the cone point. The Gauss-Bonnet theorem

applies with this notion of curvature to give ∑m

i=1ki=4π.

Thurston considers only those cone metrics which are nonnegatively curved, i.e. all ki>0.2Fix-

ing a tuple of positive real numbers ⃗

k=(k1,...,km)with each 0 <ki<2πand ∑m

i=1ki=4π,

Thurston considers the moduli space of Euclidean cone metrics on the sphere with curvatures ⃗

kup

to orientation-preserving similarity. We denote this space M(⃗

k).

There is a natural map from M(⃗

k)to (a ﬁnite cover of) the usual moduli space of conformal

structures on the punctured sphere by simply taking the conformal class of a ﬂat cone metric.

There is also an inverse map inspired by the Schwarz-Christoﬀel mapping of complex analysis.

Thurston uses this idea to show that the moduli space M(⃗

k)is actually orbifold-isomorphic to

the moduli space of conformal structures on m-punctured spheres with punctures labeled by the

ki[Thu98, Proposition 8.1]. This more classical moduli space is a complex orbifold of dimension

m−3. The orbifold fundamental group of the moduli space is Mod(S0,m;⃗

k), the group of mapping

classes of the m-punctured sphere that preserve the labeling by curvatures.

In his paper, Thurston shows directly that the moduli space of cone metrics has complex dimension

m−3 by giving local CPm−3coordinates on M(⃗

k)in terms of cocyles on the sphere with twisted/local

2A theorem of Alexandrov implies that every such metric arises uniquely as the intrinsic length metric on the

boundary of a convex polyhedron in Euclidean space.

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THEBURAUREPRESENTATIONANDSHAPESOFPOLYHEDRAETHANDLUGIEAbstract.WeuseageometricapproachtoshowthatthereducedBuraurepresentationspecial-izedatrootsofunityhasanotherincarnationasthemonodromyrepresentationofamodulispaceofEuclideanconemetricsonthesphere,asdescribedbyThurston.Usingthetheoryoforbifolds,welev...

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THE BURAU REPRESENTATION AND SHAPES OF POLYHEDRA ETHAN DLUGIE Abstract. We use a geometric approach to show that the reduced Burau representation special-

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