Twist Knot Invariants and Volume Conjecture

Chern–Simons theory provides a natural framework to construct a variety of knot invariants. The calculation of colored HOMFLY-PT polynomials of knots using SU(N) Chern–Simons theory requires the knowledge of 6j-symbols for the quantum group \(U_q(\frak {sl}_N)\) which are not known for arbitrary representation. Interestingly, our conjectured formula for superpolynomials (categorification of colored HOMFLY-PT polynomials) of twist knots led to deducing closed form expression for these symbols for a class of multiplicity-free \(U_q(\frak {sl}_N)\) representation. Using the twist knot superpolynomials, we compute the classical and quantum super-A-polynomials and test the categorified version of the quantum volume conjecture.