DAHA and skein algebra of surfaces: double-torus knots

Kazuhiro Hikami

doi:10.1007/s11005-019-01189-5

DAHA and skein algebra of surfaces: double-torus knots

Kazuhiro Hikami

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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. 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.

Original language	English
Pages (from-to)	2305-2358
Number of pages	54
Journal	Letters in Mathematical Physics
Volume	109
Issue number	10
DOIs	https://doi.org/10.1007/s11005-019-01189-5
Publication status	Published - Oct 1 2019

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11005-019-01189-5

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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. 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.",

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AB - 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.

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DAHA and skein algebra of surfaces: double-torus knots

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