Principal drivers

• Jim Propp
• Ralph Schiffler
• plus many coauthors of both
• a new notion of length ?

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

Positivity conjecture

• cluster alg. from surfaces
• general case

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