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Nimble evolution for pretzel Khovanov polynomials
ITEP, Moscow 117218, Russia.
Moscow Inst Phys & Technol, Dolgoprudnyi, Russia;ITEP, Moscow 117218, Russia;Inst Informat Transmiss Problems, Moscow 127994, Russia.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Moscow Inst Phys & Technol, Dolgoprudnyi, Russia;ITEP, Moscow 117218, Russia;Inst Informat Transmiss Problems, Moscow 127994, Russia.
2019 (English)In: European Physical Journal C, ISSN 1434-6044, E-ISSN 1434-6052, Vol. 79, no 10, article id 867Article in journal (Refereed) Published
Abstract [en]

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 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 not equal -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 =q2T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and -q2. However, in thick knots' regions extra eigenvalues emerge, and they are powers of the "naive" lambda, namely, they are equal to lambda 2,,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,,lambda g oscillate "faster" than the one of lambda. Hence, we call this type of evolution "nimble".

Place, publisher, year, edition, pages
SPRINGER , 2019. Vol. 79, no 10, article id 867
National Category
Subatomic Physics
Identifiers
URN: urn:nbn:se:uu:diva-396956DOI: 10.1140/epjc/s10052-019-7303-5ISI: 000492031700004OAI: oai:DiVA.org:uu-396956DiVA, id: diva2:1369777
Available from: 2019-11-13 Created: 2019-11-13 Last updated: 2019-11-13Bibliographically approved

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