Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman Model

TL;DR

This paper investigates algebraic invariants of the Cavendar-Farris-Neyman model on level-1 phylogenetic networks, focusing on sunlet networks, to enhance understanding of evolutionary models beyond trees.

Contribution

It identifies quadratic invariants for the CFN model on level-1 networks and conjectures these generate the entire ideal, advancing algebraic understanding of phylogenetic network models.

Findings

Determined all quadratic invariants in the sunlet network ideal.

Conjectured that these quadratic invariants generate the full ideal.

Reduced the problem to invariants of sunlet networks.

Abstract

Phylogenetic networks can model more complicated evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models that are used to model evolution on trees can also be extended to networks and similar questions, such as the identifiability of the network parameter or the invariants of the model, can be asked. In this paper we focus on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks. We do this by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We then determine all quadratic invariants in the sunlet network ideal which we conjecture generate the full ideal.