Nimble evolution for pretzel Khovanov polynomials

Abstract

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, 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 evolution is governed by the standard T-deformation lambda of the eigenvalues of the R-matrix. Emerging in the thick knots regions are additional Lyapunov exponents, which are multiples of the naive ones. Such frequency doubling is typical for non-linear dynamics, and our observation can signal about a hidden non-linearity of superpolynomial evolution. Since evolution with eigenvalues lambda^2, ..., lambda^g is "faster" than the one with lambda in the thin-knot region, we name it "nimble.

arXiv comment16 pages

References in corpus (38)

• On Khovanov's categorification of the Jones polynomial
• Khovanov-Rozansky Homology and Topological Strings
• Matrix factorizations and link homology II
• Soft matrix models and Chern-Simons partition functions
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Evolution method and "differential hierarchy" of colored knot polynomials
• Colored HOMFLY polynomials for the pretzel knots and links
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Multiplicity-free quantum 6j-symbols for U_q(sl_N)
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• On colored HOMFLY polynomials for twist knots
• Tabulating knot polynomials for arborescent knots
• Knot invariants from Virasoro related representation and pretzel knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Towards matrix model representation of HOMFLY polynomials
• Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial
• On the defect and stability of differential expansion
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• Quantum Racah matrices and 3-strand braids in irreps R with |R|=4
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials
• Tangle blocks in the theory of link invariants
• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• Are there p-adic knot invariants?
• Rectangular superpolynomials for the figure-eight knot
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Knot polynomials for twist satellites
• On Hopf-induced deformation of topological locus
• Orthogonal Polynomials in Mathematical Physics
• Factorization of differential expansion for non-rectangular representations
• The rational Khovanov homology of 3-strand pretzel links

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• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Bipartite expansion beyond biparticity
• Algebra of quantum $\mathcal{C}$-polynomials
• Evolution properties of the knot's defect
• Khovanov-Rozansky cycle calculus for bipartite links
• Towards Catastrophe theory for Khovanov-Rozansky homology
• Machine learning of the well known things
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
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