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This dissertation concerns two subjects that the author explored in his doctoral research: the Burau representation of the braid groups and moduli spaces of Euclidean cone surfaces. We present some general theory on each subject and the connection that eventually ties the two together. In particular, we see original results of the author that leverage the geometry of the moduli spaces of flat cone spheres to identify the kernel of certain evaluations of the Burau representation at roots of unity. Figures abound.

In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

We show that most of the genus-zero subgroups of the braid group $\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned: there is a very restricted class of \enquote{primitive} genus-zero subgroups such that these subgroups and their genus-zero intersections determine all the Alexander invariants. Then, we classify the primitive subgroups in a special subclass. This result implies the known classification of the dihedral covers of irreducible trigonal curves., Comment: 21 pages, 1 Table

We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which have appeared in the literature.

In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials., 20 pages, 3 figures; v2: minor changes, to appear in Proc. Edinburgh Math. Soc

We consider the graph $E_{n+1,1}$ with (n+1) generators $\sigma_1,..., \sigma_{n}$, and $\delta$, where $\sigma_{i}$ has an edge with $\sigma_{i+1}$ for $i=1,...,n+1$, and $ \sigma_{1}$ has an edge with $\delta$. We then define the Artin group of the graph $E_{n+1,1}$ for $n=3$ and $n=4$ and consider its reduced Perron's representation of degrees four and five respectively. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain necessary and sufficient conditions that guarantee the irreducibility of the representations for $n=3$ and $4$ .

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.

A long-standing open problem is to determine for which values of n the Burau representation Psi_n of the braid group B_n is faithful. Following work of Moody, Long-Paton, and Bigelow, the remaining open case is n = 4. One criterion states that Psi_n is unfaithful if and only if there exists a pair of arcs in the n-punctured disk D_n such that a certain associated polynomial is zero. In this paper, we use a computer search to show that there is no such arc-pair in D_4 with 2000 or fewer intersections, thus certifying the faithfulness of Psi_4 up to this point. We also investigate the structure of the set of arc-pair polynomials, observing a striking periodicity that holds between those that are, in some sense, 'closest' to zero. This is the first instance known to the authors of a deeper analysis of this polynomial set.

The (reduced) Burau representation V n of the braid group B n is obtained from the action of B n on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H *(B n ; V n ). Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over F q is q.

In this paper, we provide a numerical tool to study a material’s coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielsen–Thurston decomposition of the dynamical system. One can thus detect and identify clusters of space–time trajectories corresponding to coherent regions of the dynamical system by solving an eigenvalue problem. Unlike previous methods, the scalable computational complexity of our braid-based approach allows the analysis of large amounts of trajectories. Studying two-dimensional flows and their induced transport and mixing properties is key to geophysical studies of atmospheric and oceanic processes. However, one often has only sparse tracer trajectories (e.g., positions of buoys in time) to infer the overall flow geometry. Fortunately, topological methods based on the theory of braid groups have recently been proposed to extract structures from such a sparse set of trajectories by measuring their entanglement. This braid viewpoint offers sound foundations for the definition of coherent structures. Yet, there have been only limited efforts in developing practical tools that can leverage topological properties for the efficient analysis of flow structures: handling a larger number of trajectories remains computationally challenging. We contribute a new and simple computational tool to extract Lagrangian structures from sparse trajectories by noting that the eigenstructure of the Burau matrix representation of a braid of particle trajectories can be used to reveal coherent regions of the flows. Detection of clusters of space–time trajectories corresponding to coherent regions of the dynamical system can thus be achieved by solving a simple eigenvalue problem. This paper establishes the theoretical foundations behind this braid eigenanalysis approach, along with numerical validations on various flows.

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated., 35 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1804.07021

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give a "strong approximation" to the answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative to a topology induced by a naturally-occurring filtration. We expect that the methods of the paper should extend to many other representations of the braid group and perhaps ultimately inform the study of knot and link polynomials.

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4., 9 pages

We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid. Vpeljemo reducirano Burauovo reprezentacijo za mešano grupo kit na eni niti, ki predstavlja splet v lečastem prostoru in pokažemo kako izračunati Alexandrov polinom spleta neposredno iz mešane kite.

Bu tezde Örgü gruplarını üreteçleri ve ilişkileriyle betimlemeye ve Artin teoremini ispatlamaya çalıştık. Ayrıca küçük indeksli örgü grupları için bu grupların kompleks temsilleri çalışıldı. Örgü grupları, Matematikte pek çok yerde karşımıza çıkan, simetrik gruplarla birlikte anılan, sonsuz elemanlı değişmez grup ailelerinden biridir. Bu gruplar, ayrıca fizikte, özellikle kuantum teorisinde uygulama alanı bulmuştur. Bu grupların temsiller yoluyla betimlenmesi pek çok probleme konu olmuştur. Bu problemlerin pek çoğu hala açık problemdir. Biz bu tezde bu grupları Artin teoremi ile betimlemeye ve B3 örgü grubunun matris temsilleri üzerinde çalıştık. Ayrıca Artin grubunun merkezi, saf örgü grubu ve Burau temsili gibi diğer önemli konulara da değinildi. In this thesis, we decribe the Braid groups of knitters with their genarators and their relations and prove the Artin theorem. We also studied the complex representations of these groups for small indexed Braid groups. Braid groups, one of the many groups in mathematics, emerging, as related to symmetric groups, are a family of non- abelian groups with infinitely many elements.These groups also found application in physics, especially in quantum theory. Description of these groups through representations has been the subject of many problems. Many of these problems are still open problems in mathematics. In this thesis, we tried to decribe these groups by Artin theorem and also we studied two dimentioal matrix representations of the Braid group B3. We also touched upon other important issues such as the center of the Artin group, the pure Braid group, and the Burau representation.

For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\backslash G/K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_\infty$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.

We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial., 13 pages, 3 Figures

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal functor from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of "pointed Lagrangian relations" between skew-Hermitian $R$-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like functor that has been described by Donaldson. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus functor to the "Alexander functor" (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter., 36 pages; very minor changes

We consider the spin representation of Artin's braid group, which has a negative index of one and was originally given by D. D. Long and explicitly computed by J.P.Tian. In our work, we find sufficient conditions under which the complex specialization of that representation, namely $\alpha :B_{n}\to GL_{n^{2}}(\mathbb C)$, is unitary relative to a nonsingular hermitian matrix.