https://doi.org/10.1140/epjc/s10052-019-7303-5
Regular Article - Theoretical Physics
Nimble evolution for pretzel Khovanov polynomials
1
Moscow Institute for Physics and Technology, Dolgoprudny, Russia
2
ITEP, Moscow, 117218, Russia
3
Department of Physics and Astronomy, Uppsala University, Box 516, 75120, Uppsala, Sweden
4
Institute for Information Transmission Problems, Moscow, 127994, Russia
* e-mail: popolit@gmail.com
Received:
14
June
2019
Accepted:
14
September
2019
Published online:
22
October
2019
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 
. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at 
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 
, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and 
. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” 
, namely, they are equal to 
. 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 is pure phase the contributions of oscillate “faster” than the one of . Hence, we call this type of evolution “nimble”.
© The Author(s), 2019
