Knots and tropical curves

TL;DR

This paper introduces a method using Tropical Geometry to associate tropical curves with knots via their Jones polynomials, revealing connections between knot invariants, character varieties, and incompressible surfaces.

Contribution

It establishes a novel link between tropical curves and knot invariants, providing explicit computations for specific knots and supporting conjectural dualities.

Findings

Tropical curves are assigned to knots using Jones polynomials.

The tropical curve explains the relation between the AJ and Slope Conjectures.

Explicit tropical curves are computed for the 4_1, 5_2, and 6_1 knots.

Abstract

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial of the knot. We compute explicitly the tropical curve for the $4_1$, $5_2$ and $6_1$ knots and verify the above prediction.