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The colored HOMFLY polynomial is the quantum invariant of oriented links in S 3 associated with irreducible representations of the quantum group 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-Marino-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

On the Hecke algebras and the colored HOMFLY polynomial