Twisted Alexander vanishing order of knots II

TL;DR

This paper advances twisted Alexander theory by giving a precise group-theoretic criterion for TAV groups via $w(G)=1$ and $G'$ not a $p$-group, and by analyzing the TAV order $\mathcal{O}(K)$ across knots. It provides a complete list of TAV groups with order $<201$, develops practical satellite constructions to realize arbitrary TAV groups, and shows there are infinitely many hyperbolic knots realizing a given TAV group and, in many cases, as its smallest TAV group with prescribed orders such as $24,60,96,120,168$. For groups of order $pqr$, it classifies possible TAV groups, constructs associated knots (e.g., $K_{p,q,r}$), and proves the existence of infinitely many hyperbolic knots with $\mathcal{O}(K)=pqr$, highlighting a rich interplay between satellite operations and vanishing phenomena. A central-extension framework yields a concise formula $\Delta_K^{\rho_n\circ f_n}(t)=\prod_{\ell=0}^{n-1} \Delta_K^{\tilde{\rho}_1\circ f_n}(e^{2\pi i \ell/(kn)}t)$, showing that certain TAV-realization properties are stable under central extensions. The work also establishes why some small orders cannot be realized by $2$-bridge knots and connects TAV order questions to broader topics like Morse–Novikov theory and 3-manifold group representations, suggesting directions for further refinement and exploration.

Abstract

In our previous work, we introduced the notion of the twisted Alexander vanishing order of knots, defined as the order of the smallest finite group for which the corresponding twisted Alexander polynomial vanishes. In this paper, we explore several properties of this invariant in detail and present a list of twisted Alexander vanishing groups of order less than $201$.

Figures (1)

• Figure 1: Construction of $K_{p,q,r}$; $(p,q,r) = (2,3,5)$.

Theorems & Definitions (54)

• Theorem 1.1: FV13-1
• Theorem 1.2: IMS23-1
• Theorem 1.3: IMS23-1
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 2.1: IMS23-1