Markov's Conjecture on integral necklaces

TL;DR

This paper explores Markov's conjecture by translating it into a combinatorial problem involving the length spectrum of the modular torus, providing a new perspective on a classical mathematical conjecture.

Contribution

It offers a geometric reformulation of Markov's conjecture and explicitly describes the set of lengths involved in the simple length spectrum on the modular torus.

Findings

Reformulation of Markov's conjecture in geometric terms

Explicit description of the length spectrum set

New combinatorial perspective on the conjecture

Abstract

We use the geometric reformulation of Markov's uniqueness conjecture in terms of the simple length spectrum on the modular torus to rewrite the conjecture in combinatorial terms by explicitly describing this set of lengths.