Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters $$n_0, \dots , n_g$$n0,⋯,ng. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at $$T\ne -1$$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 the two eigenvalues 1 and $$\lambda = q^2 T$$λ=q2T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and $$-q^2$$-q2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” $$\lambda $$λ, namely, they are equal to $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when $$\lambda $$λ is pure phase the contributions of $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg oscillate “faster” than the one of $$\lambda $$λ. Hence, we call this type of evolution “nimble”.