H. R. Morton
It is shown how Fiedler's `small state-sum' invariant for a braid can be calculated from the 2-variable Alexander polynomial of the link which consists of the closed braid together with the braid axis.
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The crossing matrix and the Burau matrix of pure braids
该研究证明纯辫的Burau矩阵在t=1处的一阶导数等于其交叉矩阵的拉普拉斯算子，并探讨了两类矩阵衍生的共轭不变量。
A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids
本文推广了Burau表示，并针对经典辫子定义了群G_n^3的表示，该表示能够检测Burau表示的核。
It is well known that the Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define $L^2$-Burau maps and use them to compute some $L^2$-Alexander torsions of links. As an application, we prove that the $L^2$-Burau maps distinguish more braids than the Burau representation.
We show that the kernel of $Burau(4) \otimes Z_p$, the reduced Burau representation with coefficients in $Z_p$ of the 4-braid group $B_4$, consists only of pseudo-Anosov braids.
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.
In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion give finite type invariants similar to the Fiedler polynomial. I investigate how these invariants distinguish exchange related braids.
We study the kernel of the evaluated Burau representation through the braid element $σ_i σ_{i+1} σ_i$. The element is significant as a part of the standard braid relation. We establish the form of this element's image raised to the $n^{th}$ power. Interestingly, the cyclotomic polynomials arise and can be used to define the expression. The main result of this paper is that the Burau representation of the braid group of $n$ strands for $n \geq 3$ is unfaithful at any primitive root of unity, excepting the first three.
The Burau representation of the braid group is faithful for n = 4
利用前人想法证明辫群的Burau表示在n = 4时是忠实的这一定理，其推论是辫群的Jones表示在n = 4时也忠实。
A previous work of A. Conway and the author introduced $L^2$-Burau maps of braids, which are generalizations of the Burau representation whose coefficients live in a more general group ring than the one of Laurent polynomials. This same work established that the $L^2$-Burau map of a braid at the group of the braid closure yields the $L^2$-Alexander torsion of the braid closure in question, as a variant of the well-known Burau-Alexander formula. In the present paper, we generalize the previous result to $L^2$-Burau maps defined over all quotients of the group of the braid closure. The link inva…
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. The group homology $H_*(B_n;V_n)$ of braid groups with coefficients in the complexified reduced Burau representation is calculated. 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 $\mathbb{F}_q$ is exactly $q$.
Geometric Monodromy of Mixed Braid Groups and the Multivariate Burau Representation
该研究探讨混合辫群$B_{n,\mathcal{P}}$在$\mathbb{P}^1$循环分支覆盖一阶上同调的单值群作用，构造其显式张成集与埃尔米特相交形式，将其约化后与多元Burau表示建立同构关系。
We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.
We establish strong constraints on the kernel of the (reduced) Burau representation $β_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $β_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $β_4\left(g\right)$, in terms of the Garside normal form of $g\in B_4$ and a new product decomposition of positive braids. We develop cance…
Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. Virtual braids correspond naturally to virtual knots. We consider the group of virtual braids on n strings VB_n and its Burau representation, in particular we study their homological properties. We prove that the plus-construction of the classifying space of the virtual braid group on the infinite number of strings is an infinite loop space which is equivalent to a product of Q(S^0), S^1 and an infinite loop space Y. Connections with the K-functor of the integers are discussed.
We categorify the coefficients of the Burau representation matrix using elementary geometrical methods. We show the faithfulness of this categorification in the sense that it detects the trivial braid.
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 $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.
The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a…
Burau representation of braid groups and $q$-rationals
本文建立了 $q$-形变有理数理论与辫群 $\mathrm{B}_3$ 的 Burau 表示之间的联系，利用 $q$-有理数的奇点集解决了该表示的忠实复数特化分类问题，并证明了特定环面之外复数特化的忠实性。
Gassner and Burau representations over $\mathbb{Z}_p$-modules
本文研究Artin辫群的Gassner与Burau表示及其模p约化，运用拓扑方法证明Gassner表示对所有n均忠实且模p忠实，并给出Burau表示模p忠实性的新证明，展望高辫群应用。
We discuss techniques for analysing the structure of the group obtained by reducing the image of the Burau representation of the braid group modulo a prime. The main tools are a certain sesquilinear form first introduced by Squier and consideration of the action of the group on a Euclidean building.
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.
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 ultim…
We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.
We study some aspects of the geometric representation theory of the Thompson and Neretin groups, suggested by their analogies with the diffeomorphism groups of the circle. We prove that the Burau representation of the Artin braid groups extends to a mapping class group $A_T$ related to Thompson's group $T$ by a short exact sequence $B_{\infty}\hookrightarrow A_T\to T$, where $B_{\infty}$ is the infinite braid group. This {\it non-commutative} extension abelianises to a central extension $0\to \Z\to A_T/[B_{\infty},B_{\infty}]\to T\to 1$ detecting the {\it discrete} version $\bar{gv}$ of the Bo…
A simple multivariable version of the reduced Burau matrix is constructed for any braid. It is shown how the multivariable Alexander polynomial for the closure of the braid can be found directly from this matrix.
Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$
本文通过 Fox 微积分建立 Burau 表示与 Knot Floer 同调的明确联系，定义范畴化行列式的不变量，并利用量子 $\mathfrak{gl}(1 \vert 1)$ 证明其恢复辫表示。
Burau representation of the Artin braid group remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for knots and links. In the present work, we show that interesting representations of braid group could be achieved using a simple and intuitive approach, where we simply analyse the path of strands in a braid and encode the over-crossings, under-crossings or no-crossings into some parameters. More precisely, at each crossing, where, for example, the strand $i$ …
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid words with Morton-Williams-Franks bound 3 and show that closed positive braids of braid index 3 are closed positive 3-braids. For closed braids on more strings, we study the alternating links occurring. In particular we classify those of braid index 4, and show that their Morton-Wi…
Faithful Burau-like representations of some rank two Garside groups and torus knot groups
本文提出一种方法，用于生成G(n,m)群在GL₂(C[t±¹, q±¹])中的忠实表示，其中G(n,m)是Garside群，其元素的Garside正规形式可通过矩阵显式恢复。该方法构造了多个忠实表示，包括推广到G(n,m)类群的Burau表示，这些群是扭结群。
Generalisations of Hecke algebras from Loop Braid Groups
本文基于环辫群和 Burau 表示的扩张构造了 Hecke 代数的推广 $LH_n$，证明其具有有限维性、包含关系、半单商由 Pascal 三角形描述、A 型箭图等表示论性质，并分析了其与参数 $t$ 及商代数 $SP_n$ 的同构关系。
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that t…
State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unk…
Closed 4-braids and the Jones unknot conjecture
针对Jones纽结猜想中4股辫子情形的未解问题，研究发现一族以拧数为参数、可能对应非平凡纽结的双变量HOMFLY-PT多项式（其Jones多项式平凡），探究其性质并提出检验方法与多股推广思路。
In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unk…
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