Abstract
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. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at Tne -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, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and -q^2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” lambda , namely, they are equal to lambda ^2, dots , lambda ^g. 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 is pure phase the contributions of lambda ^2, dots , lambda ^g oscillate “faster” than the one of lambda . Hence, we call this type of evolution “nimble”.
Highlights
We call this type of evolution “nimble”. It is well-known that HOMFLY-PT polynomials [1,2,3,4,5,6,7,8] possess evolution structure [9,10,11,12,13,14,15,16,17,20,21]. This has a simple explanation within the modernized Reshetikhin-Turaev (MRT) formalism [22,23,24,25,26,27], and the evolution eigenvalues are those of the R-matrix in the relevant representations
In attempts to find a refined version of MRT, one can try to observe a similar structure for Khovanov polynomials empirically – and is immediately gratified: evolution was already proved to persist for the series of torus and twist knots [48,49,50,51]
In the region where all winding parameters are positive, reduced Khovanov polynomials for pretzel knots of genus g are given by the general formula
Summary
It is well-known that HOMFLY-PT polynomials [1,2,3,4,5,6,7,8] possess evolution structure [9,10,11,12,13,14,15,16,17,20,21]. 2 for a definition), which includes the entire twist and double-twist series, but only 2-strand sub-family of torus knots. Their evolution at HOMFLY-PT and, partly, superpolynomial levels was described in detail in [9,10,11,12,13,14,15,16,17,20,21] and [52,53,54,55,57].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
