Super Markov Numbers and Signed Double Dimer Covers

TL;DR

The paper develops a Z2-graded generalization of Markov numbers—Super Markov Numbers—via decorated super Teichmüller theory on the once-punctured torus and certain annuli. It presents two complementary frameworks: a matrix (holonomy) approach using products of flat Osp(1|2) matrices along Christoffel-word paths, and a combinatorial approach via signed double dimer counts on snake graphs derived from Christoffel words, yielding the key relation $SM_{p/q} = M_{p/q} + \\hat{M}_{p/q} σθ$. Central results show that $SM_{p/q}$ equals the $(1,2)$-entry of a holonomy $H$ and that, combinatorially, $SM_{p/q}$ equals the ordinary Markov number plus a signed “soul” term, with explicit calculations for several slopes. The work also extends to super annuli, derives super-integrable recurrences, and outlines rich open questions about positivity, monotonicity, and broader Diophantine analogues, linking hyperbolic geometry, cluster algebras, and super-geometry through double dimer theory. The significance lies in providing a concrete, checkable bridge between geometric structures in super Teichmüller theory and arithmetic/d combinatorial invariants, with potential applications to generalized Diophantine equations and superfrieze patterns.

Abstract

We provide a superalgebraic analogue of Markov numbers, which are defined as the Grassmann integer solutions to the equation $x^2 + y^2 + z^2 + (xy + yz + xz)ε= 3(1 + ε)xyz$, as well as applications to the Decorated Super Teichmüller spaces associated to the once-punctured torus and certain annuli. We conclude with further directions for study.

Figures (19)

• Figure 1: The default orientation of a generic triangulation where each fan segment is colored differently.
• Figure 2: (Left): Super Ptolemy Transformation. (Right): Equivalence Relation.
• Figure 3: The graph $\Gamma_T$, with $T$ in dashed lines
• Figure 4: The three types of holonomy matrices.
• Figure 5: (Left): Triangulation of a once-punctured torus with its two triangles oriented cyclically. (Middle): Result after flipping $z$ to get $z'$. (Right): Then flipping $y$ to get $y"$. The triangulation continues to be oriented cyclically.

Theorems & Definitions (46)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5: elementary steps
• Proposition 2.6
• Remark 2.7
• Definition 2.8
• Theorem 2.9
• Lemma 2.10