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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
Knot Atlas
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