Twisted Alexander invariants and hyperbolic volume. (English) Zbl 1383.57016

Goda, Hiroshi

Proc. Japan Acad., Ser. A 93, No. 7, 61-66 (2017).

From the introduction: In this note, we show that the Y. Yamaguchi’s method [On the non-acyclic Reidemeister torsion for knots. Dissertation at the Uniersity of Tokyo (2007); Ann. Inst. Fourier 58, No. 1, 337–362 (2008; Zbl 1158.57027)] is applicable to higher-dimensional Reidemeister torsion invariants, so that we have a formula of the hyperbolic volume of a knot complement using twisted Alexander invariants. Let \(\Delta_{K,\rho_n}(t)\) be the twisted Alexander invariant of N. Wada’s notation [Topology 33, No. 2, 241–256 (1994; Zbl 0822.57006)]. For the integer \(k(>1)\), set \({\mathcal A}_{K,2k}(t):= {\Delta_{K,\rho_{2k}}(t)\over \Delta_{K,\rho_2(t)}}\) and \({\mathcal A}_{K,2k+1}(t):={\Delta_{K,\rho_{2k+1}}(t)\over \Delta_{K,\rho_3}}\).
Theorem 1.1. Let \(K\) be a hyperbolic knot in the 3-sphere. Then \[ \lim_{k\to\infty}\, {\log|{\mathcal A}_{K,2k+1}(1)|\over (2k+1)^2}= \lim_{k\to\infty}\, {\log|{\mathcal A}_{K,2k}(1)\over (2k)^2}= {\text{Vol}(K)\over 4\pi}. \] In the last section, we give some calculations for the figure eight knot. The details, including link case, will be given elsewhere.

MSC:

57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57Q45	Knots and links in high dimensions (PL-topology) (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

twisted Alexander polynomial; hyperbolic knot; volume; Reidemeister torsion

Citations:

Zbl 1158.57027; Zbl 0822.57006