Ortho-integral surfaces

TL;DR

This paper develops a combinatorial framework for orthogeodesics on hyperbolic surfaces, introduces recursive trace computation, explores integer trace surfaces related to Apollonian packings, and offers a combinatorial proof of Basmajian's identity.

Contribution

It presents a novel combinatorial structure for orthogeodesics, a recursive method for trace calculation, and links to integral Apollonian packings, along with a new root-flipping concept and a combinatorial proof of Basmajian's identity.

Findings

Recursive method for orthogeodesic trace computation

Existence of surfaces with integer orthogeodesic traces

Connection between orthogeodesics and Apollonian circle packings

Abstract

This paper introduces a combinatorial structure of orthogeodesics on hyperbolic surfaces and presents several relations among them. As a primary application, we propose a recursive method for computing the trace (the hyperbolic cosine of the length) of orthogeodesics and establish the existence of surfaces where the trace of each orthogeodesic is an integer. These surfaces and their orthogeodesics are closely related to integral Apollonian circle packings. Notably, we found a new type of root-flipping that transitions between roots in different quadratic Diophantine equations of a certain type, with Vieta root-flipping as a special case. Finally, we provide a combinatorial proof of Basmajian's identity for hyperbolic surfaces, akin to Bowditch's combinatorial proof of the McShane identity.