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

In [Kim] for an oriented surface $S_{g}$ of genus $g$ it is shown that links in $S_{g} \times S^{1}$ can be presented by virtual diagrams with a decoration, so called, 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 $VB_{n}^{dl}$. Alexander and Markov theorem for links in $S_{g}\times S^{1}$ can be proved analogously to the work in [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 Iwahori-Hecke algebra.
\begin{thebibliography} {100} \bibitem{Kim} S. Kim, {\it The Groups $G_{n}^{k}$ with additional structures,} Matematicheskie Zametki, Vol. 103, No. 4 (2018), pp. 549 – 567. \bibitem{NegiPrabhakarKamada} K. Negi, M. Prabhakar, S. Kamada, Twisted virtual braids and twisted links, Osaka J. Math. 61(4): 569-590 (October 2024). \end{thebibliography}