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Towards effective topological field theory for knots. (English) Zbl 1331.81264
Summary: Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when ”fingers” and ”propagators” are substituting $$\mathcal{R}$$-matrices in arbitrary closed braids with $$m$$-strands. Original version of [A. Anokhina, the authors et al., ibid. 882, 171–194 (2014: Zbl 1285.81035)] corresponds to the case $$m = 2$$, and our generalization sheds additional light on the structure of those mysterious formulas. Explicit expressions are now combined from Racah matrices of the type $$R \otimes R \otimes \overline{R} \longrightarrow \overline{R}$$ and mixing matrices in the sectors $$R^{\otimes 3} \longrightarrow Q$$. Further extension is provided by composition rules, allowing to glue two blocks, connected by an $$m$$-strand braid (they generalize the product formula for ordinary composite knots with $$m = 1$$).

##### MSC:
 81T45 Topological field theories in quantum mechanics 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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