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arXiv · 1904.10277

Nimble evolution for pretzel Khovanov polynomials

Aleksandra Anokhina·Alexei Morozov·Aleksandr Popolitov
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Abstract

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots evolution is governed by the standard T-deformation lambda of the eigenvalues of the R-matrix. Emerging in the thick knots regions are additional Lyapunov exponents, which are multiples of the naive ones. Such frequency doubling is typical for non-linear dynamics, and our observation can signal about a hidden non-linearity of superpolynomial evolution. Since evolution with eigenvalues lambda^2, ..., lambda^g is "faster" than the one with lambda in the thin-knot region, we name it "nimble.

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Aleksandra Anokhina, Alexei Morozov, Aleksandr Popolitov. 2019-09-20. Nimble evolution for pretzel Khovanov polynomials. https://doi.org/10.1140/epjc%2Fs10052-019-7303-5

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