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- Title
Nimble evolution for pretzel Khovanov polynomials.
- Authors
Anokhina, Aleksandra; Morozov, Alexei; Popolitov, Aleksandr
- 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 , ⋯ , 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 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 λ = 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" λ , namely, they are equal to λ 2 , ⋯ , λ 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 λ is pure phase the contributions of λ 2 , ⋯ , λ g oscillate "faster" than the one of λ . Hence, we call this type of evolution "nimble".
- Subjects
KNOT theory; LAURENT series; POLYNOMIALS; SECOND harmonic generation; PRETZELS; BIOLOGICAL evolution; COMPLEX variables
- Publication
European Physical Journal C -- Particles & Fields, 2019, Vol 79, Issue 10, pN.PAG
- ISSN
1434-6044
- Publication type
Article
- DOI
10.1140/epjc/s10052-019-7303-5