I will discuss the direct evaluation of knot invariants using the
connection between Chern-Simons theory and Wess-Zumino conformal field
theory. From these pool of invariants, I will present how we attempted
to write closed form expression for $SU(N)$ quantum racah coefficients
involving a symmetric and their conjugate representations. I will
briefly introduce Khovanov's approach of obtaining homological
invariants. Finally, I will give some idea on recent activities in
this mathematical physics area: (i) obtaining superpolynomials,
A-polynomials (ii) understanding these polynomials from the BPS states
in string theory, refined Chern-Simons theory and 3-dimensional
gravity partition functions.
