On the Hecke algebras and the colored HOMFLY polynomial

The colored HOMFLY polynomial is the quantum invariant of oriented links in S 3 S^3 associated with irreducible representations of the quantum group U q ( s l N ) U_q(sl_N) . In this paper, using an approach to calculate quantum invariants of links via the cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.