On the Hecke algebras and the colored HOMFLY polynomial

Lin, Xiao-Song; Zheng, Hao

doi:10.1090/S0002-9947-09-04691-1

On the Hecke algebras and the colored HOMFLY polynomial
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by Xiao-Song Lin and Hao Zheng PDF

Trans. Amer. Math. Soc. 362 (2010), 1-18 Request permission

Abstract:

The colored HOMFLY polynomial is the quantum invariant of oriented links in $S^3$ associated with irreducible representations of the quantum group $U_q(\mathrm {sl}_N)$. 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.

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