# Matchings and snakes

greg mc

May 2021

## length

\((p,q)\) curve

- \(\sqrt p^2 + q^2\)
- \(|p| + |q|\)
- \(\ell_w\)
- complexity of another object ?

A matching of a graph G is a set of pairwise non-adjacent edges, none of which are loops.

A perfect matching matches all vertices of the graph.

## Matching polynomial

5 perfect matchings

- \(x^4 + y^4 + x^2z^2 + 2x^2y^2\)
- match \(\rightarrow\) monomial=product of labels

13 perfect matchings

src

## Snake graph from primitive

\(A=(1,1)\), \(B=(2,1)\), \(C=(3,2)\)

## Positivity conjecture

- cluster algebras
- seed is like a Markoff triple \((x,y,z)\)
- mutation is like a Vieta flip
- Markoff number \(\rightarrow\) Laurent polynomial
- proof (idea) : coeffs are coeffs of the snake polynomial

## Calculus of arcs/snake graphs

- sophistocated combinatorial proof
- relations to continued fractions
- looking for applications

## Markoff numbers

\(A,B \in \mathbb{Z}^2\)

\(|AB|=\) number perfect matchings for associated snake graph

src

## Exercise to prove

Prove the relation for matchings of triples of snake graphs

\(tr ab + tr ab^{-1} = (tr a) (tr b)\)

## Ptolemy inequality