We study a topological aspect of rank-1 double-affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. C∨C1-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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics