Confirming Brennan's conjecture numerically on a counterexample to Thurston's $K=2$ conjecture

TL;DR

This paper numerically verifies that a known counterexample to Thurston's $K=2$ conjecture does not serve as a counterexample to Brennan's conjecture, supporting the conjecture's validity in that context.

Contribution

It provides the first numerical analysis confirming that Epstein, Marden, and Marković's counterexample does not violate Brennan's conjecture.

Findings

The counterexample to Thurston's $K=2$ conjecture does not contradict Brennan's conjecture.

Numerical evidence supports Brennan's conjecture in the context of the counterexample.

The study bridges the gap between theoretical conjectures and numerical validation.

Abstract

It was shown by Bishop that if Thurston's $K = 2$ conjecture holds for some planar domain, then Brennan's conjecture holds for the Riemann map of that domain as well. In this paper we show numerically that the original counterexample to Thurston's $K=2$ conjecture given by Epstein, Marden and Markovi\'{c} is not a counterexample to Brennan's conjecture.