# Univalent Polynomials and Hubbard Trees

## TL;DR

This paper classifies certain rational functions univalent outside the unit disk with maximal cusps and double points, establishing a correspondence with anti-holomorphic polynomials via combinatorial bi-angled trees.

## Contribution

It introduces bi-angled trees to classify these rational functions and proves a one-to-one correspondence with anti-holomorphic polynomials' Hubbard trees.

## Key findings

- Bi-angled trees are realizable by such rational functions.
- Each rational function is essentially uniquely determined by its bi-angled tree.
- The classification links these functions to anti-holomorphic polynomials with fixed critical points.

## Abstract

We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natural 1:1 correspondence with anti-holomorphic polynomials of degree $d$ with $d-1$ distinct, fixed critical points (classified by their Hubbard trees).