TL;DR
This paper introduces a new approach to studying generalized Markov numbers via semigroups of matrices and bipartite graph matchings, linking algebraic, combinatorial, and geometric perspectives.
Contribution
It presents a novel method for computing Markov numbers using bipartite graphs called wug-snake graphs, connecting matrix semigroups to graph theory and number geometry.
Findings
Generalized Markov numbers can be obtained from counting perfect matchings in wug-snake graphs.
The construction links semigroup theory with geometric number theory and classical Markov minima.
A new combinatorial framework for understanding Markov numbers is established.
Abstract
In this paper, we systematically study generalized Markov numbers arising from semigroups of reduced integer matrices. This construction allows us to find these numbers by counting perfect matchings of a new family of bipartite graphs, which we call wug-snake graphs. We also show how this relates to the geometry of numbers and the classical theory of Markov minima.
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