Abstract
Knot polynomials colored with symmetric representations of SLq(N) satisfy difference equations as functions of representation parameter, which look like quantization of classical \( \mathcal{A} \)-polynomials. However, they are quite difficult to derive and investigate. Much simpler should be the equations for coefficients of differential expansion nicknamed quantum \( \mathcal{C} \)-polynomials. It turns out that, for each knot, one can actually derive two difference equations of a finite order for these coefficients, those with shifts in spin n of the representation and in A = qN. Thus, the \( \mathcal{C} \)-polynomials are much richer and form an entire ring. We demonstrate this with the examples of various defect zero knots, mostly discussing the entire twist family.
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Mironov, A., Morozov, A. Algebra of quantum \( \mathcal{C} \)-polynomials. J. High Energ. Phys. 2021, 142 (2021). https://doi.org/10.1007/JHEP02(2021)142
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DOI: https://doi.org/10.1007/JHEP02(2021)142