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 $S^{ 2}\times S^{ 1}$.
Categories: math.GT math.AG