Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
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Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams Masaya Kameyama1 · Satoshi Nawata2,3,4 Hao Derrick Zhang2
· Runkai Tao2 ·
Received: 11 February 2019 / Revised: 18 June 2020 / Accepted: 21 July 2020 © Springer Nature B.V. 2020
Abstract We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot. Keywords Knot invariants · Special functions Mathematics Subject Classification 57K14 · 33D52
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . Convention . . . . . . . . . . . . . . . . . . . . . . . . . 2 Colored HOMFLY-PT polynomials . . . . . . . . . . . . 3 Poincaré polynomials of colored HOMFLY-PT homology
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Satoshi Nawata [email protected] Masaya Kameyama [email protected] Runkai Tao [email protected] Hao Derrick Zhang [email protected]
1
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
2
Department of Physics and Center for Field Theory and Particle Physics, Fudan University, 2005 Songhu Road, Shanghai 200438, China
3
Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
4
Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA
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M. Kameyama et al. 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Interpolation Macdonald polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction Colored HOMFLY-PT polynomials are two-variable quantum knot invariants associated with irreducible representations of A N type Lie algebras. Although several methods to compute HOMFLY-PT polynomials for arbitrary color in principle are known, carrying out explicit computation is practically very challenging for general non-torus knots and colors. However, in recent years, studying the structural properties of colored HOMFLY-PT polynomials, closed-form expressions for symmetric representations have been found for a certain class of non-torus knots. Actually, the structural properties become more apparent at the level of HOMFLYPT homology that categorifies quantum HOMFLY-PT polynomials. Lately, the HOMFLY-PT homology colored by arbitrary representations has been defined in [1]. Although it is formidable to carry out computation of homology via the definition, various structural properties of HOMFLY-PT homology have been uncovered by combining mathematical definitions and physical predictions. In particular, it was propo
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