Computing tropical curves via homotopy continuation

TL;DR

This paper introduces a numerical algebraic geometry method to compute tropical curves by leveraging amoeba connections, with an application to determining Newton polygons of knot A-polynomials.

Contribution

It presents a novel numerical approach for computing tropical curves based on amoeba connections, including an implementation and application to knot invariants.

Findings

Successfully computed tropical curves using the proposed method

Demonstrated the technique's effectiveness on Newton polygons of A-polynomials

Provided an implementation that can be used for further research

Abstract

Exploiting a connection between amoebas and tropical curves, we devise a method for computing tropical curves using numerical algebraic geometry and give an implementation. As an application, we use this technique to compute Newton polygons of $A$-polynomials of knots.