Tabulation of prime knots in lens spaces

TL;DR

This paper computationally catalogs prime knots with up to five crossings in lens spaces and the solid torus, analyzing their skein modules and hyperbolic structures to distinguish knot types.

Contribution

It provides the first comprehensive tabulation of prime knots in lens spaces up to five crossings, including skein module calculations and hyperbolic structure comparisons.

Findings

Most knots are distinguished by skein modules.

Some knots require hyperbolic invariants for distinction.

Limited data for certain cases with p ≥ 13.

Abstract

Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces $L(p,q)$. For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein modules. For the handful of cases where the skein modules fail to detect inequivalent knots, we calculate and compare the hyperbolic structures of the knot complements. We were unable to resolve a handful of 5-crossing cases for $p\geq 13$.