# Nimble evolution for pretzel Khovanov polynomials

## TL;DR

This paper proposes explicit evolution formulas for Khovanov polynomials of pretzel knots, revealing complex boundary behaviors and non-linear dynamics, especially distinguishing thin and thick knot regions.

## Contribution

It introduces a comprehensive description of Khovanov polynomial evolution for pretzel knots, including boundary phenomena and the discovery of Lyapunov exponents indicating non-linear dynamics.

## Key findings

- Evolution formulas for pretzel knots are provided for genera 1 and 2.
- Boundary regions exhibit abrupt changes in polynomial evolution.
- Thick knots show additional Lyapunov exponents, indicating non-linear behavior.

## 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.