# DAHA and skein algebra on surface: double-torus knots

## TL;DR

This paper explores the connection between double affine Hecke algebras (DAHA) and skein algebras on surfaces, constructing representations for genus-two surfaces and proposing a DAHA polynomial for double-torus knots, linking it to colored Jones polynomials.

## Contribution

It introduces a novel DAHA representation for skein algebras on genus-two surfaces and proposes a new DAHA polynomial for double-torus knots, connecting algebraic and topological invariants.

## Key findings

- Established relationship between DAHA of A1 and CC1 types and skein algebras on punctured surfaces.
- Constructed DAHA representation for skein algebra on genus-two surface.
- Proposed DAHA polynomial for double-torus knots and related it to colored Jones polynomial.

## Abstract

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus two Heegaard surface in S3. Discussed is a relationship between the DAHA polynomial and the colored Jones polynomial.