Snake graphs and continued fractions

TL;DR

This paper explores the combinatorial representation of continued fractions using snake graphs, linking them to number theory concepts like sums of squares and Markov numbers.

Contribution

It introduces a new combinatorial framework connecting snake graphs with continued fractions, Euclidean algorithm, and number theory applications.

Findings

Representation of continued fractions via snake graphs

Connections to Euclidean division and convergents

Applications to sums of squares and Markov numbers

Abstract

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.