Finiteness of the image of the Reidemeister torsion of a splice

Abstract

The set RT(M) of values of the SL(2, C)-Reidemeister torsion of a 3-manifold M can be both finite and infinite. We prove that RT(M) is a finite set if M is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and A-polynomials of knots.

An algebraic property of Reidemeister torsion

“…Here, 𝑓 𝐶 (𝐿, 𝑀) ∈ ℤ[𝐿, 𝑀] is the polynomial 16 − 𝐿((𝑀 32 + 1) − 4(𝑀 30 + 𝑀 2 ) − 2(𝑀 28 + 𝑀 4 ) + 16(𝑀 26 + 𝑀 6 ) + 13(𝑀 24 + 𝑀 8 ) − 32(𝑀 22 + 𝑀 10 ) − 46(𝑀 20 + 𝑀 12 ) + 20(𝑀 18 + 𝑀 14 ) + 70𝑀 16 ) + 𝑀 16 , which is derived from 𝑟(𝐶) ⊂ 𝑋(𝜕𝐸(𝐾 0 )). Note that the condition on 𝐾 in Corollary 1.4 is generic enough.…”

“…Note that the knot $$K_0$$ is alternating, hyperbolic, and fibered (see Section 5). Theorem 1.3 is motivated by the authors' previous work [14]. They investigated the set $$\begin{equation*} \mathit {RT}(M)=\lbrace \tau _\rho (M) \mid [\rho ] \in X^\mathrm{irr}(M),\ \text{acyclic}\rbrace \subset \mathbb {C} \end{equation*}$$of all values of the Reidemeister torsion for irreducible representations.…”

The adjoint Reidemeister torsion for the connected sum of knots

“…It is known that the character variety X (M ) has a component of dimension ≥ 2 as the connected sum admits a so-called bending. We refer to [JM87, PP13,KN20] for details on the bending construction.…”

An algebraic property of Reidemeister torsion

“…Note that the knot K 0 is alternating, hyperbolic, and fibered (see Section 4). Theorem 1.3 is motivated by the authors' previous work [13]. They investigated the set…”

“…of all values of the Reidemeister torsion for irreducible representations. In [13], the authors proved that for 2-bridge knots K 1 and K 2 , the set…”