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

Seongjeong Kim

Paper

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

of genus

links in

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

. We denote the group of braids with double lines by

. The Alexander and Markov theorems for links in

can be proved analogously to the work in \cite{NegiPrabhakarKamada}. We show that if we restrict our interest to the group

generated by braids with double lines, but without virtual crossings, then the Hecke algebra of

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