17 research outputs found
Special colored Superpolynomials and their representation-dependence
We introduce the notion of "special superpolynomials" by putting q=1 in the
formulas for reduced superpolynomials. In this way we obtain a generalization
of special HOMFLY polynomials depending on one extra parameter t. Special
HOMFLY are known to depend on representation R in especially simple way: as
|R|-th power of the fundamental ones. We show that the same dependence persists
for our special superpolynomials in the case of symmetric representations, at
least for the 2-strand torus and figure-eight knots. For antisymmetric
representations the same is true, but for t=1 and arbitrary q. It would be
interesting to find an interpolation between these two relations for arbitrary
representations, but no superpolynomails are yet available in this case.Comment: 5 page
Super-A-polynomials for Twist Knots
We conjecture formulae of the colored superpolynomials for a class of twist
knots 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.Comment: 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 adde