Skeins on tori

TL;DR

This paper studies the algebraic structures of skein invariants of 3- and 2-tori for certain groups, revealing connections to double affine Hecke algebras and providing explicit formulas and isomorphisms.

Contribution

It introduces new formulas for skein module dimensions and establishes isomorphisms between skein algebras and DAHA, advancing understanding of skein theory on tori.

Findings

Dimension formulas for skein modules of T^3.

Isomorphism between N-point skein algebra and DAHA.

Surjective homomorphisms from DAHA to skein algebras.

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.