Nimble evolution for pretzel Khovanov polynomials
Authors: Aleksandra Anokhina, Alexei Morozov and Aleksandr Popolitov
Preprint number: UUITP-15/19
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 λ 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 λ2, . . . , λg is ”faster” than the one with λ in the thin-knot region, we name it “nimble”.