Braids for Knots in $S_{g} \times S^{1}$ and the affine Hecke algebra
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915678650368000 |
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| author | Kim, Seongjeong |
| author_facet | Kim, Seongjeong |
| contents | 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 $S^{2}\times S^{1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Braids for Knots in $S_{g} \times S^{1}$ and the affine Hecke algebra Kim, Seongjeong Geometric Topology Algebraic Geometry 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 $S^{2}\times S^{1}$. |
| title | Braids for Knots in $S_{g} \times S^{1}$ and the affine Hecke algebra |
| topic | Geometric Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2512.14027 |