Publication Date:
2019
Description:
〈h3〉Abstract〈/h3〉
〈p〉We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables 〈em〉q〈/em〉 and 〈em〉T〈/em〉, for pretzel knots of genus 〈em〉g〈/em〉 in some regions in the space of winding parameters 〈span〉
〈span〉\(n_0, \dots , n_g\)〈/span〉
〈/span〉. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at 〈span〉
〈span〉\(T\ne -1\)〈/span〉
〈/span〉 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 〈span〉
〈span〉\(\lambda = q^2 T\)〈/span〉
〈/span〉, governing the evolution, are the standard 〈em〉T〈/em〉-deformation of the eigenvalues of the 〈em〉R〈/em〉-matrix 1 and 〈span〉
〈span〉\(-q^2\)〈/span〉
〈/span〉. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” 〈span〉
〈span〉\(\lambda \)〈/span〉
〈/span〉, namely, they are equal to 〈span〉
〈span〉\(\lambda ^2, \dots , \lambda ^g\)〈/span〉
〈/span〉. 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 〈span〉
〈span〉\(\lambda \)〈/span〉
〈/span〉 is pure phase the contributions of 〈span〉
〈span〉\(\lambda ^2, \dots , \lambda ^g\)〈/span〉
〈/span〉 oscillate “faster” than the one of 〈span〉
〈span〉\(\lambda \)〈/span〉
〈/span〉. Hence, we call this type of evolution “nimble”.〈/p〉
Print ISSN:
1434-6044
Electronic ISSN:
1434-6052
Topics:
Physics
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