Statement of research

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1 Neil J. Fullarton, October Introduction My research interests lie in the fields of geometric group theory and low-dimensional topology. In particular, I study the topological, geometric, and combinatorial properties of mapping class groups of oriented surfaces, and automorphism groups of free and right-angled Artin groups. Mapping classes of surfaces and automorphisms of free groups have been studied for well over a century, initially by Dehn and Magnus, and then by Thurston and others, due to the major role these objects play throughout mathematics. More recently, right-angled Artin groups have risen to prominence due to ground-breaking work of Agol [1] and Wise [46] on 3-manifolds. My current research projects concern: Cohomology of mapping class groups and their finite index subgroups ( 2). The period mapping on Culler Vogtmann outer space, and its relationship to the homology of free group automorphisms in the Torelli group ( 3). Geometric models for the automorphism group of a right-angled Artin group, and the topological and dynamical information they contain ( 4). Branched coverings of graphs and manifolds, and a Birman Hilden theorem for homotopy equivalences compatible with such coverings ( 3). Faithfulness of the Burau representation of the braid group B 4 ( 5). The mapping class group Mod g of the closed, oriented surface Σ g of genus g is the group of orientation-preserving homeomorphisms of Σ g, modulo homotopy. This group holds influence in a remarkable number of mathematical fields, from algebraic geometry and complex analysis to the theory of 3- and 4-manifolds [30]. For instance, Mod g is the orbifold fundamental group of M g, the moduli space of genus g Riemann surfaces, an object of central importance in the algebro-geometric world. The group Out(F n ) of outer automorphisms of the free group F n is the group of automorphisms of F n modulo inner automorphisms. Viewing Out(F n ) topologically as the mapping class group of a graph, we observe many similarities with Mod g. In particular, Out(F n ) is the orbifold fundamental group of G n, the moduli space of rank n metric graphs, and plays the same role for tropical geometers as Mod g does for algebraic geometers [21]. More algebraically, Out(F n ) may be thought of as the group of symmetries of the free group F n. Given the ubiquity of free groups throughout mathematics, we should strive to understand their symmetries as fully as possible. Free and free abelian groups are examples of right-angled Artin groups ( RAAGs ), which are groups whose only defining relators are commutators between generators. These groups allow us to interpolate between the well-studied groups F n and Z n, and, passing to their automorphism groups, between Out(F n ) and Out(Z n ) = GL n (Z). RAAGs have a strong 1

2 geometric flavor, with close ties to hyperbolic 3-manifolds and CAT(0) geometry [22], while more fundamentally, their elementary definition leads them to have a widespread presence throughout mathematics. One driving force behind my research program is seeking to strengthen the robust analogies that hold between the triad of groups formed by Mod g, Out(F n ) and GL n (Z). These analogies exist because of the close relationship between the groups π 1 (Σ g ), F n and Z n, whose automorphisms form the groups in this triad. These analogies are not merely descriptive: one often fruitful approach to problems is to translate useful tools and objects from one setting to another. For instance, the success of Teichmüller space in studying Mod g led to the defining of Culler Vogtmann s outer space [27], which has been essential in pushing forward our understanding of Out(F n ). The remainder of this document highlights the main components of my research program, giving details of specific problems, my progress to date, and the direction of future work. 2. Cohomology of mapping class groups A fundamental open question about the mapping class group is: which of the rational cohomology groups of Mod g are non-zero? An initial step toward answering this was made by Harer [36], who computed the virtual cohomological dimension ( vcd ) of Mod g to be 4g 5, meaning that Mod g has no rational cohomology above this dimension. More recently, Church Farb Putman [26] showed that at this top possible dimension, we have H 4g 5 (Mod g, Q) = 0. With Putman, I have found a wealth of finite index subgroups of Mod g whose rational cohomology groups are highly non-zero at this top dimension, answering a question of Looijenga. Let Mod g [l] denote the principal level l congruence subgroup of Mod g, i.e. the kernel of the action of Mod g on H 1 (Σ g ; Z/l). Theorem 2.1 (Fullarton Putman [33]). Let g, l 2, and let p be a prime factor of l. The Q-vector space H 4g 5 (Mod g [l]; Q) has dimension at least Sp 2g (Z/p) g(p 2g 1), where Sp 2g (Z/p) denotes the group of (2g) (2g) symplectic matrices with entries in the finite field Z/p. In particular, this lower bound is a polynomial in p with leading term 1 g p(2g 2 ). Our theorem has a notable consequence for the coherent cohomological dimension of moduli space, M g. This is the maximum value i for which there exists a coherent sheaf F on M g such that the (sheaf) cohomology group H i (M g ; F) is non-zero [37]. In general, this dimension captures important geometric information about a variety, such as whether it is affine or not. Corollary 2.2. The coherent cohomological dimension of the moduli space M g of Riemann surfaces is at least g 2. This lower bound is known to be sharp for 2 g 5, and a conjecture of Looijenga would imply that it is sharp for all g. We prove Theorem 2.1 in two main steps: 2

3 (i) First, we use a duality theorem of Bieri Eckmann to identify H 4g 5 (Mod g [l]; Q) with the homology group H 0 (Mod g [l]; St g ), where St g is the Steinberg module of Σ g. This module is the top dimensional homology group of the curve complex of Σ g, a combinatorial model capturing intersection information between simple closed curves in Σ g. (ii) The above duality thus reduces the problem to understanding the action of Mod g [l] on St g. We do this by constructing a novel homomorphism that maps St g to a quotient of a similarly defined module associated to the Tits building of a finite vector space. Future work. All of the objects and theorems used in the proof of Theorem 2.1 have welldefined analogs for Out(F n ), so the above proof schematic could, in principle, be leveraged to obtain similar results regarding the cohomology of Out(F n ) and finite index subgroups. However, there are several obstacles to overcome. Such calculations would be particularly interesting, as recent work of Bartholdi [5] showed that Out(F 7 ) has non-zero rational cohomology at its vcd. This provides a stark, unexpected contrast with the Church Farb Putman zero result for Mod g, which requires explanation. The main impediment to such work is the lack of an appropriate description of the Steinberg module for F n. Bestvina Feighn [7] showed that Out(F n ) satisfies a homological duality theorem, however their Steinberg module does not yield to the sorts of attack that St g does. The first step in such a program for Out(F n ) is thus: Problem 2.3. Obtain a description of Bestvina Feighn s Steinberg module more compatible with the proof techniques of Theorem 2.1. With such a description in place, I plan to investigate the following questions. A principal congruence subgroup of Out(F n ) is the kernel of the action of Out(F n ) on H 1 (F n ; Z/l) for some l 2. Question 2.4. Is the rational cohomology group of Out(F n ) non-zero at its (virtual) cohomological dimension, 2n 3? Question 2.5. Do the principal congruence subgroups of Out(F n ) have non-zero rational cohomology at their cohomological dimension? Answers to these questions would form part of a larger research program in geometric group theory of striving to understand the cohomology of the group Out(F n ). 3. The geometry of free group automorphisms Culler Vogtmann s outer space CV n consists of metric graphs Γ of rank n with a marking, i.e. a choice of generating set for π 1 (Γ) = F n, up to conjugation [27]. By changing these markings, the group Out(F n ) acts on the contractible space CV n with finite stabilizers, producing the quotient G n, the moduli space of rank n metric graphs. This action on CV n enables us to extract geometric information about the group Out(F n ). For instance, the rational homology groups of Out(F n ) and CV n are naturally identified through this action. Outer space CV n is a free group analog of two other well-studied spaces: from the geometric world, the Teichmüller space of marked, metric surfaces of fixed genus, and from the arithmetic world, the homogeneous space Q n of rank n positive definite quadratic forms. The connection between CV n and Q n is more apparent when Q n is identified with the space of marked, flat metrics on the n-torus. 3

4 The group Out(F n ) may succumb to techniques of linear algebra, as discussed later, via the classical epimorphism Out(F n ) GL n (Z), induced by abelianizing F n. However, this surjection has a large, enigmatic kernel I n called the Torelli group, about which there are still many open questions. A more tractable subgroup of I n is the partial conjugation subgroup, whose elements map the standard generators of F n to conjugates of themselves. The period mapping. Given any marked, metric graph Γ representing a point in CV n, there is a natural way to define an inner product on H 1 (Γ; R): the marking gives a basis of cycles for this homology group, and the inner product of two such cycles is computed by summing (with signs and multiplicity) the lengths of the edges they both traverse [4]. Converting this inner product to a quadratic form, we obtain the period mapping Φ : CV n Q n. Since the definition of Φ only involves a choice of homology basis for H 1 (Γ; R), it factors through the Torelli space quotient T n of CV n by the Torelli group I n. The definition of the period mapping Φ is inspired by that of the classical period mapping ρ from algebraic geometry. The map ρ determines much of the algebraic structure of a Riemann surface (c.f. the Torelli theorem and Abel s theorem [3]), so we are well-motivated to consider its free group analog Φ. A natural question to consider is: what do the fibers of Φ look like? Together with Bregman, I have characterized the topological structure of these fibers. This generalizes work of Baker [4], who explicitly described the period mapping s fibers in the case n = 3. Theorem 3.1 (Bregman Fullarton [14]). The fibers of Φ in Torelli space are aspherical, π 1 -injective subspaces. Moreover, their fundamental groups are conjugate into the partial conjugation subgroup of I n. A topological space X is aspherical if its only non-trivial homotopy group is π 1 (X). Such spaces form the foundation of a topological analysis of discrete groups, since two aspherical spaces are homotopy equivalent if and only if their fundamental groups are isomorphic [19]. This important fact allows us to associate the topological properties of X to the group π 1 (X). Theorem 3.1 thus suggests that the period mapping is detecting the geometry of the Torelli group and its partial conjugation subgroup, and should be further studied. Bregman and I also observed that Φ behaves as a 2-to-1 branched covered on a large subspace of T n. We isolated a subspace of T n of hyperelliptic graphs, denoted L n, which acts as a branch locus. The fundamental group of L n is isomorphic to a subgroup of I n called the hyperelliptic Torelli group, ST (n). Basic examples of elements of this subgroup are doubled commutator transvections and separating π-twists. Together with Bregman, I proved that the former suffice to generate. Theorem 3.2 (Bregman Fullarton [14]). The hyperelliptic Torelli group generated by doubled commutator transvections. The proof of Theorem 3.2 builds upon my previous work on the palindromic Torelli subgroup PI n of Aut(F n ), which is an Aut(F n ) version of the hyperelliptic Torelli group. Theorem 3.2 verifies, in the Out(F n ) setting, a conjecture of Brendle regarding generating PI n. In this previous work, I proved the following. Theorem 3.3 (Fullarton [31]). Let n 3. 4

5 (1) The palindromic Torelli group PI n is generated by the set of doubled commutator transvections and separating π-twists. (2) The set of doubled commutator transvections generates a proper subgroup of PI n. This generating set for PI n yields a particularly nice finite presentation for the principal level 2 congruence group, ker(gl n (Z) GL(n, Z/2)). Corollary 3.4 (Fullarton [31]). The principal level 2 congruence subgroup of GL n (Z) has a finite presentation with relations similar to those appearing in the Steinberg presentation for SL(n, Z). Thomas and I built upon these results, defining the palindromic Torelli group and, more generally, palindromic automorphisms for right-angled Artin groups [35]. We obtained an analog of Theorem 3.3 in the RAAG setting, which is the first step in generalizing the study of the period mapping into the realm of right-angled Artin groups. Theorem 3.2 is a free group analog of a generating set for the hyperelliptic Torelli subgroup of Mod g, obtained by Brendle Margalit Putman [16]. Their generating set, together with an observation of Hain, was used to show that the hyperelliptic locus of marked Riemann surfaces becomes simply-connected when certain surfaces of degenerate type are adjoined (c.f. the Deligne Mumford compactification of M g [28]). Margalit asked the following question. Question 3.5 (Margalit). Is there a free group analog of the hyperelliptic locus of Riemann surfaces? Does it become simply-connected when certain points at infinity are added? Bregman and I gave a positive answer to this question. The branching set L n is a very natural analog of the classical hyperelliptic locus, and we proved the following theorem. Theorem 3.6 (Bregman Fullarton [14]). The locus L n of hyperelliptic graphs becomes simply-connected when hyperelliptic graphs of rank less than n are adjoined. Building upon these initial results, I will continue to probe the structure of the period mapping and its branch locus L n. Bestvina Bux Margalit [6] determined a sharp bound for the dimension of Torelli space T n, and also used T n to find infinitely many independent cycles in the top dimensional homology group of π 1 (T n ) = I n. Their techniques are compatible with a decomposition of the fibers of Φ that we used to prove Theorem 3.1, so we will approach the following problem. By Theorem 3.1, this problem corresponds to studying the homology of the partial conjugation subgroup in Out(F n ). Problem 3.7. Use the methods of Bestvina Bux Margalit to construct independent cycles in the homology of the period mapping fibers. We will also further explore the hyperelliptic Torelli group ST (n). We will use the locus L n to compute the cohomological dimension of ST (n), which is known to either be n 1 or n 2. Because of these bounds, a promising first step would be to compute this dimension for ST (3). Since a group is free if and only if its cohomological dimension is 1, I conjecture the following. Conjecture 3.8 (Fullarton). The group ST (3) is an infinite rank free group. This would compare favorably with the mapping class group setting: a theorem of Mess [42] establishes that the smallest non-trivial Torelli group (the genus 2 case), which equals the hyperelliptic Torelli group of Mod 2, is an infinite rank free group. 5

6 A Birman Hilden theorem for free groups. The hyperelliptic automorphisms in Out(F n ) discussed previously in this section centralize, by definition, a certain order 2 element of Out(F n ). The group of such automorphisms is a free group analog of the hyperelliptic mapping class group SMod g of Σ g, which is defined to be the centralizer of a certain involution of Σ g. The group SMod g is studied using the Birman Hilden theorem [12], which we now describe, with the goal of translating the theorem into the free group setting. Let p : Σ g Σ h be a (possibly branched) regular cover of surfaces. The celebrated Birman Hilden theorem asserts that a subgroup of mapping classes of Mod h may be lifted through the cover, and is isomorphic to a p-symmetric subgroup of mapping classes of Σ g modulo the deck group of the cover. This theorem not only provides a wealth of interesting homomorphisms between different mapping class groups, but also allows Mod g to be studied by passing to surfaces of simpler topological type. For instance, the first finite presentation known for any mapping class group was of Mod 2, and it was obtained by applying the Birman Hilden theorem to Mod 2 viewed as a branched double cover of a punctured sphere [11]. Following this success in the surface context, Margalit posed the following question. Question 3.9 (Margalit). Does a Birman Hilden theorem hold for the group Out(F n )? In forthcoming work, together with Calabrese and Winarski, I have obtained such an analog of the Birman Hilden theorem, for free groups. An initial difficulty was finding the right notion of (branched) cover for free groups. Mere branched covers of graphs, a tempting first notion, do not retain enough information to be of use. We thus generalized the class of objects we consider to orbifolds. Loosely speaking, these are branched covers of spaces that record how each branch point is stabilized under the covering map. In order to state our Birman Hilden theorem, let p : X Y be an orbifold covering map, where X is a finite graph or manifold, and p is induced by a suitably geometric action of some group G on X, subject to some mild conditions. Let H(X) denote the group of homotopy classes of homotopy equivalences of X, and let LH(Y ) denote the subgroup of H(Y ) of such homotopy classes with a representative that lifts to X. These groups are related as follows. Theorem 3.10 (Calabrese Fullarton Winarski [20]). The group of liftable homotopy equivalences LH(Y ) is isomorphic to the normalizer of G in H(X), modulo G itself. Weaker versions of this theorem were known previously to Rose [44] and Krstić [40]. The Birman Hilden viewpoint for free groups has already proved fruitful: Bregman and I used Rose s weaker version of Theorem 3.10 in our analysis of hyperelliptic automorphisms of F n. Calabrese, Winarski and I will use our above theorem to generate new examples of virtual injections between Out(F n ) and Out(F m ) for n m (a line of research initiated by Bogopolski Puga [13] and Bridson Vogtmann [18], and continued by Kielak [39]). Problem Produce new examples of (virtual) injections from Out(F n ) to Out(F m ) (for some n m). We will also investigate the effect of lifting upon the homotopy equivalences of Y. For instance, Aramayona Leininger Souto [2] showed that it is possible for a pseudo-anosov mapping class of a surface to map under an injection to another mapping class group as a product of Dehn twists. This is a surprising phenomenon, and it is natural to wonder if its 6

7 occurs for Out(F n ), for fully irreducible or atoroidal automorphisms, both of which are free group analog of a pseudo-anosov mapping class. Question 3.12 (Fullarton). Do there exist covers of graphs through which fully irreducible or atoroidal elements of Out(F n ) lift to automorphisms of a simpler type (e.g. a product of disjoint transvections)? What properties cause this to happen? 4. Automorphisms of right-angled Artin groups Given a finite, simplicial graph Γ, we define the right-angled Artin group ( RAAG ) A Γ via the presentation that has a generator v i for each vertex of Γ, and a relation v i v j = v j v i for each edge (v i, v j ) in Γ. RAAGs are generalizations of free and free abelian groups; moreover, they allow us to interpolate between these important classes of groups, by varying the number of edges in Γ. RAAGs have risen to prominence in recent years due to their important role in Agol s solutions to the long-standing virtually Haken and virtually fibering conjectures in the theory of hyperbolic 3-manifolds [1]. They appear in this setting because there is a particularly nice CAT(0) cube complex, the RAAG s so-called Salvetti complex, that is a K(A Γ, 1) space. Salvetti complexes act like universal special cube complexes, which were a key tool in work of Agol and Wise [46]. Algebraic ridigity of Out(A Γ ). The groups Out(F n ) and GL n (Z) both satisfy an algebraic rigidity property [38], [17], since for any n, the outer automorphism group of the former is trivial, while that of the latter has order at most 4. It is natural to ask if this rigidity persists for the outer automorphism group of any A Γ. The following two theorems assess this finiteness. Theorem 4.1 (Fullarton [32]). For any k N, there exist finite graphs Γ and such that Out(Out(A Γ )) and Out(Aut(A )) contain finite subgroups of order at least k. Theorem 4.2 (Bregman Fullarton [15]). For any k N, there exists a finite graph Γ such that Out(Out(A Γ )) contains PGL(k, Z), and hence is infinite. These theorems are a first step towards understanding the abstract commensurator Comm(Out(A Γ )). For any group G, its abstract commensurator is a vast generalization of the automorphism group of G: Comm(G) is the group of all isomorphisms between finite index subgroups of G, up to a natural equivalence. The algebraic ridigity of Out(F n ) and GL n (Z) is also seen in their commensurators [29], with Comm(Out(F n )) = Out(F n ) and Comm(GL n (Z)) = GL(n, Q). I plan to investigate the behaviour of Comm(Out(A Γ )) for an arbitrary RAAG. Problem 4.3. Calculate the abstract commensurator of Out(A Γ ) for arbitrary A Γ. Outer space for RAAGs. As with Out(F n ) and GL n (Z), we expect Out(A Γ ) to be the orbifold fundamental group of some moduli space of geometric structures. The universal cover of this moduli space, in analogy with Culler Vogtmann outer space CV n for Out(F n ), and the space of marked, flat tori Q n for GL n (Z), would allow us to investigate geometric properties of Out(A Γ ). We refer to this universal cover as an outer space for RAAGs. Certain classes (two-dimensional [23] and untwisted [25] RAAGs) have good notions of an outer space. Most recently, Charney Stambaugh Vogtmann developed a space K Γ consisting of marked blow-ups of Salvetti complexes. These are cube complexes homotopy 7

8 equivalent to the Salvetti complex S Γ of A Γ via certain permissible collapses of cube subcomplexes. The space K Γ is contractible, and a large subgroup Out (A Γ ) of Out(A Γ ) acts on it, with finite stabilizers. By allowing the edge lengths in these blow-ups to vary, to form rectangular complexes, we obtain a higher dimensional space L Γ, which deformation retracts onto K Γ. It appears challenging to extend the definition of K Γ to obtain a space on which the full group Out(A Γ ) acts on, as it is not clear how to enlarge the class of permissible blow-ups of S Γ in a controlled way. Drawing inspiration from other complexes on which Out(F n ) acts, I will take a different approach. Culler Vogtmann s CV n embeds into a simplicial complex known as the sphere complex S n as a dense, open subset. The sphere complex is defined as follows: a regular neighborhood in R 3 of a wedge of n circles is doubled along its boundary to yield a closed 3-manifold M n with fundamental group F n. The vertices in S n are isotopy classes of embedded 2-spheres in M n, with simplices being spanned by collections of pairwise disjoint 2-spheres. The above construction of M n can be mimicked for a non-free A Γ by, for instance, doubling a neighborhood of S Γ in some high-dimensional Euclidean space. We may then consider collections of disjoint spheres to build a sphere complex using this manifold. Problem 4.4. Construct a sphere complex for an arbitrary A Γ, and establish its properties (e.g. contractibility, an action of Out(A Γ ) upon it). The advantage of this approach is that it circumvents the need to generalize the definition of Salvetti blow-up, but still yields an outer space -like object. Indeed, the sphere complex s definition may clarify which blow-ups of Salvetti complexes should be permitted. There is another description of the sphere complex S n in terms of decomposition of F n into free splittings. Charney has proposed a similar complex for a RAAG, using a natural generalization of a free splitting to an Artin splitting, although this complex has never appeared in the literature. I expect this complex of Charney to also play a role in determining a RAAG s outer space. Knowledge of the spaces K Γ and L Γ themselves is also currently rather limited. A good description of a compactification of L Γ could initiate the study of the dynamics of the action of Out (A Γ ). RAAGs whose defining graph Γ are triangle-free and connected have had their outer space L Γ compactified, by Vijayan [45], building upon work of Charney Margolis [24]. However, the case for a general RAAG is completely open. Problem 4.5. Compactify the space L Γ in a manner that generalizes the compactification of CV n. Given such a compactification, the following problem would be approachable. Problem 4.6. Classify elements of Out (A Γ ) according to their dynamics on the boundary of this compactification. A solution to this problem would generalize results in this area on the dynamics of elements in Out(F n ). 5. Computations in braid groups The n-strand braid group B n appears frequently throughout mathematics, in large part due to its close relationship with the motion of sets of n marked points in the disk D n R 2. 8

9 Braids give rise to knots and links in S 3 [10], appear in subgroups of mapping class groups as products of Dehn twists, and even play a role in certain monoidal categories. Braid groups enjoy many attractive properties, such as finite-presentability and linearity. Linearity was established using the faithful Lawrence Krammer representation. This representation generalizes the Burau representation Ψ n of B n [8], which is constructed using an action of B n on the infinite cyclic cover of the marked disk D n. The question of faithfulness of Ψ n was open for decades, but was answered, by work of Moody [43], Long Paton [41] and Bigelow [8], for all values of n except 4: Ψ n is faithful for n = 2, 3 but is not if n 5. R. Shadrach and I investigated the remaining open case, n = 4. As remarked by Bigelow [9], a non-trivial braid in the kernel of Ψ 4 would yield a probable candidate for proving that the Jones polynomial cannot detect the unknot. The Burau representation s lack of faithfulness for large n was shown using various criteria involving certain pairs A of arcs in D n intersecting an even number of times. Such a pair (α, β) defines a Burau polynomial, P (α, β). One criterion for faithfulness says that Ψ n is not faithful if and only if there exists an essentially intersecting arc-pair (α, β) A such that P (α, β) = 0. Shadrach and I certify the faithfulness of Ψ 4 up to pairs of arc intersecting a large number of times, as stated below. Theorem 5.1 (Fullarton Shadrach [34]). Any pair of arc (α, β) A intersecting at most 2000 times has non-zero Burau polynomial. We prove this using a computer search. We describe arcs using weighted train tracks, and use the action of B 4 on arcs to calculate a finite yet sufficient number of Burau polynomials at each intersection number. Having found no zero polynomials, we turned to a qualitative analysis of the Burau polynomials we calculated. The norm of a polynomial is the sum of its coefficients absolute values. We calculated the minimum norm at each intersection number, and observed a striking periodicity. Theorem 5.2 (Fullarton Shadrach [34]). Let 2k [44, 500]. If 2k is a multiple of 6, then the minimum norm over all Burau polynomials arising from an arc-pair with 2k intersections is 10. Otherwise, the minimum norm is 8. This periodicity almost certainly persists beyond intersection number 500; due to limits on computation time, we were only able to check up to this bound. Note that if Ψ 4 is unfaithful, there must be some large intersection number at which the periodicity ceases. Future work. In proving Theorem 5.2, we observed that arc-pairs appear to fall into natural families, whose Burau polynomials maybe be obtained from each other in a controlled way. In particular, in all known families, one arc-pair s polynomial is non-zero if and only if all the family s polynomials are. This suggests that a sparser search through the set of arc-pairs is possible. Problem 5.3. Batch arc-pairs in D n into the families suggested by Theorem 5.2 in order to carry out a sparse search for a zero Burau polynomial. This would allow us to explore beyond the bound of 2000 stated in Theorem 5.1 for zero Burau polynomials. 9

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