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-polynomials 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 Apolynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

3d-3d correspondence revisited

“…Poincaré polynomials for infinite families of twist knots derived in [23,26] share analogous features. It becomes clear now that various properties of trefoil and figure-8 knots, discussed earlier, should also be present for other knots, such as thin knots discussed above.…”

“…This structure can be clearly seen in the example of (2, 2p + 1) torus knots considered in [23,26], whose (normalized) colored superpolynomials take the form…”

“…If the uncolored superpolynomial on the right hand side is a sum of a few terms, its r'th power can be written as a (multiple) summation involving Newton binomials, which for arbitrary q turn out to be replaced by q-binomials [23,26]. This structure can be clearly seen in the example of (2, 2p + 1) torus knots considered in [23,26], whose (normalized) colored superpolynomials take the form…”

“…those whose superpolynomials are found in [23,26]), and to Poincaré polynomials of other homological invariants. In general, the required integrals will not be one-dimensional, but will require higher-dimensional integration cycles.…”

Factorization of differential expansion for antiparallel double-braid knots

“…Modern group theory is incapable to provide the answers beyond pure symmetric and antisymmetric representations [97][98][99][100] [103] and [104] for some inclusive Racah matrices for R = [3,1] and R = [2, 2] respectively). Further progress on these lines seems to be beyond the current computer capacities.…”