Dihedral Linking Invariants

TL;DR

This paper introduces an algorithm to compute dihedral linking invariants for all odd p-colored knots, expanding previous methods and providing extensive tabulations for numerous knots, aiding knot classification.

Contribution

It generalizes Perko's algorithm to all odd p, enabling systematic computation of dihedral linking invariants for p-colored knots.

Findings

Algorithm successfully computes invariants for thousands of knots.

Provides comprehensive tables of dihedral linking invariants.

Enhances understanding of knot invariants and their applications.

Abstract

A Fox p-colored knot $K$ in $S^3$ gives rise to a $p$-fold branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko, and tabulate the invariant for thousands of $p$-colorable knots.