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Differential hierarchy and additional grading of knot polynomials

S. B. Arthamonov^a, A. D. Mironov^ab, A. Yu. Morozov<sup>a

a</sup> Institute for Theoretical and Experimental Physics, Moscow, Russia
^b Lebedev Physics Institute, RAS, Moscow, Russia

Abstract: Colored knot polynomials have a special $Z$-expansion in certain combinations of differentials, which depend on the representation. The expansion coefficients are functions of three variables $A$, $q$, and $t$ and can be regarded as new distinguished coordinates on the space of knot polynomials, analogous to the coefficients of the alternative character expansion. These new variables decompose especially simply when the representation is embedded into a product of fundamental representations. The recently proposed fourth grading is seemingly a simple redefinition of these new coordinates, elegant, but in no way distinguished. If this is so, then it does not provide any new independent knot invariants, but it can instead be regarded as one more piece of evidence in support of a hidden differential hierarchy $(Z$-expansion{)} structure behind the knot polynomials.

Keywords: Chern–Simons theory, colored knot invariant, superpolynomial.

Received: 11.12.2013

DOI: 10.4213/tmf8625

 English version: Theoretical and Mathematical Physics, 2014, 179:2, 509–542

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