{"title":"Skeins on tori","authors":"Sam Gunningham, David Jordan, Monica Vazirani","doi":"arxiv-2409.05613","DOIUrl":null,"url":null,"abstract":"We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the\ntwo-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum\nparameter. We obtain formulas for the dimension of the skein module of $T^3$,\nand we describe the algebraic structure of the skein category of $T^2$ --\nnamely of the $n$-point relative skein algebras. The case $n=N$ (the Schur-Weyl case) is special in our analysis. We construct\nan isomorphism between the $N$-point relative skein algebra and the double\naffine Hecke algebra at specialized parameters. As a consequence, we prove that\nall tangles in the relative $N$-point skein algebra are in fact equivalent to\nlinear combinations of braids, modulo skein relations. More generally for $n$\nan integer multiple of $N$, we construct a surjective homomorphism from an\nappropriate DAHA to the $n$-point relative skein algebra. In the case $G=SL_2$ corresponding to the Kauffman bracket we give proofs\ndirectly using skein relations. Our analysis of skein categories in higher rank\nhinges instead on the combinatorics of multisegment representations when\nrestricting from DAHA to AHA and nonvanishing properties of parabolic sign\nidempotents upon them.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Quantum Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05613","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the
two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum
parameter. We obtain formulas for the dimension of the skein module of $T^3$,
and we describe the algebraic structure of the skein category of $T^2$ --
namely of the $n$-point relative skein algebras. The case $n=N$ (the Schur-Weyl case) is special in our analysis. We construct
an isomorphism between the $N$-point relative skein algebra and the double
affine Hecke algebra at specialized parameters. As a consequence, we prove that
all tangles in the relative $N$-point skein algebra are in fact equivalent to
linear combinations of braids, modulo skein relations. More generally for $n$
an integer multiple of $N$, we construct a surjective homomorphism from an
appropriate DAHA to the $n$-point relative skein algebra. In the case $G=SL_2$ corresponding to the Kauffman bracket we give proofs
directly using skein relations. Our analysis of skein categories in higher rank
hinges instead on the combinatorics of multisegment representations when
restricting from DAHA to AHA and nonvanishing properties of parabolic sign
idempotents upon them.