# Problem on Mutant Pairs of Hyperbolic Polyhedra

## TL;DR

This paper introduces a mutation concept for hyperbolic polyhedra, explores their commensurability, and highlights the need for new techniques due to the limitations of existing methods used for knots.

## Contribution

It proposes a mutation framework for hyperbolic polyhedra and raises open questions about their commensurability, emphasizing the challenge of analyzing compact polyhedra.

## Key findings

- Examples of mutant pairs with unknown commensurability
- Highlighting the limitations of cusp-based techniques for compact polyhedra
- Call for development of new methods to study mutant pairs

## Abstract

We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we consider are compact, so techniques involving cusps that are typically used to distinguishing mutant pairs of knots are not applicable. Indeed, new techniques may need to be developed to study commensurability of mutant pairs of polyhedra.