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

TL;DR

This paper introduces a simplified algorithm for computing twisted Alexander polynomials of knots, enabling the authors to distinguish non-slice knots and classify mutant pretzel knots in knot concordance.

Contribution

The authors develop a faster method to compute twisted polynomials, leading to new results in knot slicing and mutation classification.

Findings

Identified 16 non-slice knots among 18 algebraically slice knots with ≤12 crossings.

Proved that mutants of P(3,7,9,11,15) are distinct concordance classes.

Provided a more efficient algorithm for twisted polynomial computation.

Abstract

Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted 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.