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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the Hecke algebras and the colored HOMFLY polynomial


Authors: Xiao-Song Lin and Hao Zheng
Journal: Trans. Amer. Math. Soc. 362 (2010), 1-18
MSC (2000): Primary 57M27, 20C08
Published electronically: July 31, 2009
MathSciNet review: 2550143
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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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Additional Information

Xiao-Song Lin
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Hao Zheng
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, People’s Republic of China
Email: zhenghao@mail.sysu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04691-1
PII: S 0002-9947(09)04691-1
Received by editor(s): August 4, 2006
Published electronically: July 31, 2009
Additional Notes: The first author was supported in part by NSF grants DMS-0404511
The second author was supported in part by an NSFC grant
Article copyright: © Copyright 2009 American Mathematical Society