Twisted Alexander vanishing order of knots

TL;DR

The paper introduces the twisted Alexander vanishing (TAV) order $\mathcal{O}(K)$ for knots, built on Friedl–Vidussi's vanishing theorem. It proves a universal lower bound $\mathcal{O}(K)\ge 24$, establishes unboundedness on non-fibered knots, and derives structural rules for connected sums, epimorphisms, periodicity, and degree-one maps, while computing $\mathcal{O}(K)$ for prime knots with $\le 10$ crossings. A complete group-theoretic characterization shows that a finite group $G$ is a TAV group if and only if it is normally generated by a single element and $[G,G]$ is not a $p$-group, with a lifting-criterion underpinning the vanishing. The work blends explicit knot computations (e.g., $K=9_{35},9_{46},10_{166}$) with lifting theory and ideals in group rings to map which groups yield vanishing twisted Alexander polynomials. It also outlines extensive computational bounds and open problems for TAV orders in knots and potential extensions to $3$-manifold groups.

Abstract

Based on a vanishing theorem for non-fibered knots due to Friedl and Vidussi, we define the twisted Alexander vanishing order of a knot to be the order of the smallest finite group such that the corresponding twisted Alexander polynomial is zero. In this paper, we show its basic properties, and provide several explicit values for knots with $10$ or fewer crossings. Moreover, we characterize a finite group admitting the zero-twisted Alexander polynomial.

Figures (1)

• Figure 1: The $(3,2)$-cable of the figure-eight knot

Theorems & Definitions (34)

• Theorem 1.1: FV13-1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Proposition 2.5