Finite orthogonal groups and periodicity of links

Maciej Borodzik; Przemysław Grabowski; Adam Król; Maria Marchwicka

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arXiv · 1810.03881

Finite orthogonal groups and periodicity of links

Maciej Borodzik·Przemysław Grabowski·Adam Król·Maria Marchwicka

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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.

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Maciej Borodzik, Przemysław Grabowski, Adam Król, Maria Marchwicka. 2018-10-09. Finite orthogonal groups and periodicity of links. https://arxiv.org/abs/1810.03881

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