Linking forms, finite orthogonal groups and periodicity of links

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.