Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young TableauxReport as inadecuate

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Advances in High Energy PhysicsVolume 2013 2013, Article ID 931830, 12 pages

Review Article

MIPT, 9 Institutsky Per., Dolgoprudny 141700, Russia

ITEP, 25 Bol. Cheremushkinskaya, Moscow 117259, Russia

Lebedev Physics Institute, 53 Leninsky Pr., Moscow 119991, Russia

Moscow State University, GSP-1, Moscow 119991, Russia

Received 22 May 2013; Accepted 7 September 2013

Academic Editor: Kadayam S. Viswanathan

Copyright © 2013 A. Anokhina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


If a knot is represented by an -strand braid, then HOMFLY polynomial in representation is a sum over characters in all representations . Coefficients in this sum are traces of products of quantum -matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity of in . If is the fundamental representation , then is equal to the number of paths in representation graph, which lead from the fundamental vertex to the vertex . In the basis of paths the entries of the relevant -matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary -matrices consist of just and blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation , then the braid has as many as strands; have a bigger size , but only paths passing through the vertex are included into the sums over paths which define the products and traces of the relevant -matrices. In the case of , this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.

Author: A. Anokhina, A. Mironov, A. Morozov, and And. Morozov



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