Publication Date:
2013-10-24
Description:
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.
Print ISSN:
1687-7357
Electronic ISSN:
1687-7365
Topics:
Physics
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