Nimble evolution for pretzel Khovanov polynomials

• Published in: The European physical journal. C, Particles and fields Particles and fields, 2019-10, Vol.79 (10), p.1-13, Article 867
• Main Authors: Anokhina, Aleksandra, Morozov, Alexei, Popolitov, Aleksandr
• Format: Article
• Language: English
• Subjects: Astronomy; Astrophysics and Cosmology; Complex variables; Eigenvalues; Elementary Particles; Evolution; Hadrons; Heavy Ions; Knots; Linearity; Logarithms; Measurement Science and Instrumentation; Multiplication; Nonlinear dynamics; Nonlinearity; Nuclear Energy; Nuclear Physics; Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Regular Article - Theoretical Physics; Second harmonic generation; Snack foods; String Theory; Switches
• ISSN: 1434-6044; 1434-6052; 1434-6052
• DOI: 10.1140/epjc/s10052-019-7303-5

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”.