Every Markov triple has a unique q-deformed polynomial solution to the q-Markov equation, and these polynomials count weighted perfect matchings of snake graphs.
Birman, Braids, links, and mapping class groups
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On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges
Every Markov triple has a unique q-deformed polynomial solution to the q-Markov equation, and these polynomials count weighted perfect matchings of snake graphs.