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Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux. (English) Zbl 1328.81123
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 \(Q\in R^{\oplus m}\). Coefficients in this sum are traces of products of quantum \(\widehat{\mathscr{R}}\)-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity \(M_{QR}\) of \(Q\) in \(R^{\oplus m}\). If \(R\) is the fundamental representation \(R=[1]=\square\), then \(M_{\square Q}\) is equal to the number of paths in representation graph, which lead from the fundamental vertex \(\square\) to the vertex \(Q\). In the basis of paths the entries of the \(m-1\) relevant \(\widehat{\mathscr{R}}\)-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 \(\widehat{\mathscr{R}}\)-matrices consist of just \(1\times1\) and \(2\times2\) 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 \(\widehat{\mathscr{R}}\)-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.

MSC:
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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