Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux

Abstract EN

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.