# Embeddability of Kimura 3ST Markov matrices

## TL;DR

This paper characterizes when Kimura 3ST Markov matrices can be embedded, computes their volume within all Markov matrices, and discusses mutation rate identifiability and symmetry issues.

## Contribution

It provides a eigenvalue-based characterization of embeddability and analyzes mutation rate identifiability in Kimura 3ST models.

## Key findings

- Eigenvalue conditions determine embeddability.
- Computed volume of embeddable matrices within all Markov matrices.
- Mutation rates are not always identifiable from substitution probabilities.

## Abstract

In this note, we characterize the embeddability of generic Kimura 3ST Markov matrices in terms of their eigenvalues. As a consequence, we are able to compute the volume of such matrices relative to the volume of all Markov matrices within the model. We also provide examples showing that, in general, mutation rates are not identifiable from substitution probabilities. These examples also illustrate that symmetries between mutation probabilities do not necessarily arise from symmetries between the corresponding mutation rates.