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Extensions of braid group representations to the monoid of singular braids

Valeriy G. Bardakov, Nafaa Chbili, Tatyana A. Kozlovskaya

Abstract

Given a representation $\varphi \colon B_n \to G_n$ of the braid group $B_n$, $n \geq 2$ into a group $G_n$, we are considering the problem of whether it is possible to extend this representation to a representation $Φ\colon SM_n \to A_n$, where $SM_n$ is the singular braid monoid and $A_n$ is an associative algebra, in which the group of units contains $G_n$. We also investigate the possibility of extending the representation $Φ\colon SM_n \to A_n$ to a representation $\widetildeΦ \colon SB_n \to A_n$ of the singular braid group $SB_n$. On the other hand, given two linear representations $\varphi_1, \varphi_2 \colon H \to GL_m(\Bbbk)$ of a group $H$ into a general linear group over a field $\Bbbk$, we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of $SB_n$ which is an extension of the Lawrence-Krammer-Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence-Krammer-Bigelow representation.

Extensions of braid group representations to the monoid of singular braids

Abstract

Given a representation of the braid group , into a group , we are considering the problem of whether it is possible to extend this representation to a representation , where is the singular braid monoid and is an associative algebra, in which the group of units contains . We also investigate the possibility of extending the representation to a representation of the singular braid group . On the other hand, given two linear representations of a group into a general linear group over a field , we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of which is an extension of the Lawrence-Krammer-Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence-Krammer-Bigelow representation.
Paper Structure (11 sections, 7 theorems, 80 equations, 2 figures)

This paper contains 11 sections, 7 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Extension of the Lawrence-Krammer-Bigelow representation
  4. Burau representation
  5. Extensions of the braid group representations
  6. Comparing LKBR and the exterior square of Burau representation
  7. Tensor product of two Burau representations
  8. Open problems and directions for further research
  9. From the Lawrence-Bigelow-Krammer representation to knot invariants
  10. Extensions of the Artin representations
  11. Representation into the Temperley--Lieb algebra.

Key Result

Theorem 3.2

Let $\varphi_{LKB} \colon B_n \longrightarrow GL(V_{m})$ be the Lawrence-Krammer-Bigelow representation and $u, v \in \mathbb{C}$. Then the map which is defined on the generators by the formulas defines a representation of $SM_n$ which is an extension of the LKBR of $B_n$. If all $\Phi^{u,v}_{LKB}(\tau_{i})$ are invertible, then we get a representation of the group $SB_n$. Moreover, for $n=3, 4

Figures (2)

  • Figure 1: The elementary braids $\sigma_i$, $\sigma_i^{-1}$ and the unit $e$.
  • Figure 2: The elementary singular braid $\tau_i$.

Theorems & Definitions (19)

  • Example 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 4.1
  • proof
  • Definition 5.1
  • Proposition 5.2
  • ...and 9 more