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

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 $S^{2}\times S^{1}$.