The (p, q)-arithmetic hyperbolic lattices; p, q greater than or equal to 6

TL;DR

This paper classifies all 16 arithmetic lattices in hyperbolic 3-space generated by two elements of finite orders p and q (both at least six) and confirms a related conjecture about orbifold Dehn surgeries on two-bridge knots.

Contribution

It provides a complete classification of (p, q)-arithmetic hyperbolic lattices with p, q ≥ 6 and verifies a conjecture on orbifold Dehn surgeries.

Findings

Exactly 16 such arithmetic lattices exist.

Confirmed the conjecture on singular set orders in orbifold Dehn surgeries.

Enhanced understanding of hyperbolic 3-space lattices and orbifold structures.

Abstract

We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos concerning the orders of the singular sets of arithmetic orbifold Dehn surgeries on two bridge knot and link complements.