Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q∈R⊗m. Coefficients in this sum are traces of products of quantum ℛ^-matrices along the braid, but these matrices act in the space of intertwiners, and their size...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2013-01-01
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| Series: | Advances in High Energy Physics |
| Online Access: | http://dx.doi.org/10.1155/2013/931830 |
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| Summary: | If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q∈R⊗m. Coefficients in this sum are traces of products of quantum ℛ^-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity MRQ of Q in R⊗m. If R is the fundamental representation R=[1]=□, then M□Q is equal to the number of paths in representation graph, which lead from the fundamental vertex □ to the vertex Q. In the basis of paths the entries of the m-1 relevant ℛ^-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 ℛ^-matrices consist of just 1×1 and 2×2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation R, then the braid has as many as m|R| strands; Q have a bigger size m|R|, but only paths passing through the vertex R are included into the sums over paths which define the products and traces of the m|R|-1 relevant ℛ^-matrices. In the case of SU(N), 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. |
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| ISSN: | 1687-7357 1687-7365 |