### Abstract

We study a topological aspect of rank-1 double-affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A_{1}-type (resp. C^{∨}C_{1}-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 S^{3}. Discussed is a relationship between the DAHA polynomial and the colored Jones polynomial.

Original language | English |
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Pages (from-to) | 2305-2358 |

Number of pages | 54 |

Journal | Letters in Mathematical Physics |

Volume | 109 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**DAHA and skein algebra of surfaces : double-torus knots.** / Hikami, Kazuhiro.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 109, no. 10, pp. 2305-2358. https://doi.org/10.1007/s11005-019-01189-5

}

TY - JOUR

T1 - DAHA and skein algebra of surfaces

T2 - double-torus knots

AU - Hikami, Kazuhiro

PY - 2019/10/1

Y1 - 2019/10/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85068189676&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068189676&partnerID=8YFLogxK

U2 - 10.1007/s11005-019-01189-5

DO - 10.1007/s11005-019-01189-5

M3 - Article

AN - SCOPUS:85068189676

VL - 109

SP - 2305

EP - 2358

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 10

ER -