Super-A-polynomials for Twist Knots

TL;DR

This paper proposes conjectural formulas for colored superpolynomials of twist knots, verifies them through various checks, and derives classical and quantum super-A-polynomials, supporting conjectures in knot theory and contact homology.

Contribution

It introduces new conjectural formulas for superpolynomials of twist knots and derives associated super-A-polynomials, providing evidence for related conjectures.

Findings

Formulas for colored superpolynomials of twist knots

Derivation of classical and quantum super-A-polynomials

Support for generalized volume and quantum volume conjectures

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.