Super-A-polynomials for Twist Knots

Abstract

We conjecture formulae of the colored superpolynomials for a class of twist knots $K_p$ where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed A-polynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

22+16 pages, 16 tables and 5 figures; with a Maple program by Xinyu Sun and a Mathematica notebook in the ancillary files linked on the right; v2 change in appendix B, typos corrected and references added; v3 change in section 3.3; v4 corrections in Ooguri-Vafa polynomials and quantum super-A-polynomials for 7_2 and 8_1 are added

References in corpus (3)

• The Refined Topological Vertex
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Super-A-polynomial for knots and BPS states

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• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Sequencing BPS Spectra
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• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• 3d-3d Correspondence Revisited
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• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
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• Link polynomial calculus and the AENV conjecture
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• The colored HOMFLYPT function is $q$-holonomic
• BPS counting for knots and combinatorics on words
• Cabling procedure for the colored HOMFLY polynomials
• Refined Chern-Simons Theory in Genus Two
• Eigenvalue hypothesis for multi-strand braids
• Special colored Superpolynomials and their representation-dependence
• Rectangular superpolynomials for the figure-eight knot
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Towards tangle calculus for Khovanov polynomials
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• Musings on SVD and pseudo entanglement entropies
• On explicit formulae of LMOV invariants
• On knots, complements, and 6j-symbols