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Colored HOMFLY polynomials for the pretzel knots and links. (English) Zbl 1388.57012
Summary: With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations \([r]\) for a huge family of (generalized) pretzel links, which are made from \(g + 1\) two strand braids, parallel or antiparallel, and depend on \(g + 1\) integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of \(g + 1\) elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled \(\mathrm{SU}_q(N)\) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations \(R\) is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary \(g\) and \(R\).

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
Software:
Knot Atlas
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