Finite orthogonal groups and periodicity of links

TL;DR

This paper investigates the existence of specific isometries on modules over rings of integers modulo prime powers and applies these results to refine criteria for the periodicity of links in three-dimensional space, making the criteria more concrete and computational.

Contribution

It provides a complete characterization of certain isometries on modules over $Z_{p^k}$ and refines Naik's periodicity criterion for links, making it effectively computable.

Findings

Complete classification of isometries of order $q^r$ with no fixed points

Refined, effectively computable periodicity criterion for links in $S^3$

Concrete restrictions for periodicity of low-crossing knots

Abstract

For a prime number $q\neq 2$ and $r>0$ we study, whether there exists an isometry of order $q^r$ acting on a free $\mathbb{Z}_{p^k}$-module equipped with a scalar product. We investigate, whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if $p\neq 2,q$. As an application we refine Naik's criterion for periodicity of links in $S^3$. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.