Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

By Christopher Herald, Paul A. Kirk, Charles Livingston. 2008

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Given a knot complement X and its p-fold cyclic cover Xp→ X, we identify twisted polynomials associated to $${GL_1\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π1(Xp) with twisted polynomials associated to related $${GL_p\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π1(X) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3, 7, 9, 11, 15), corresponding to permutations of (7, 9, 11, 15), represent distinct concordance classes.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a2874d37fc6adf91b61da5773b0692cb1d529c41