Linking forms, finite orthogonal groups and periodicity of links

Abstract

For a prime number 𝑞 ≠ 2 and 𝑟 > 0, we study whether there exists an isometry of order 𝑞^𝑟 acting on a free 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 ≠ 2,𝑞. As an application, we refine Naik's criterion for periodicity of links in 𝑆³. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.