Braids for Knots in $S_{g} \times S^{1}$ and the affine Hecke algebra

TL;DR

This paper introduces braids with double lines for links in $S_{g} imes S^{1}$, establishes their algebraic properties, and connects the affine Hecke algebra to the Kauffman bracket skein module, extending knot theory in complex 3-manifolds.

Contribution

It defines braids with double lines, proves Alexander and Markov theorems for these links, and links the Hecke algebra to the affine Hecke algebra and skein modules.

Findings

Hecke algebra of braids with double lines is isomorphic to the affine Hecke algebra.

Established Alexander and Markov theorems for links in $S_{g} imes S^{1}$.

Defined a Markov trace from the affine Hecke algebra to the skein module.

Abstract

In \cite{Kim} it is shown that for an oriented surface $S_{g}$ of genus $g$ links in $S_{g} \times S^{1}$ can be presented by virtual diagrams with a decoration, called {\em double lines}. In this paper, first we define braids with double lines for links in $S_{g}\times S^{1}$. We denote the group of braids with double lines by $VB_{n}^{dl}$. The Alexander and Markov theorems for links in $S_{g}\times S^{1}$ can be proved analogously to the work in \cite{NegiPrabhakarKamada}. We show that if we restrict our interest to the group $B_{n}^{dl}$ generated by braids with double lines, but without virtual crossings, then the Hecke algebra of $B_{n}^{dl}$ is isomorphic to the affine Hecke algebra. Moreover, we define a Markov trace from the affine Hecke algebra to the Kauffman bracket skein module of $S^{2}\times S^{1}$.