Non-Abelian Reidemeister Torsion for Twist Knots

Abstract

This paper gives an explicit formula for the SL 2 (C)−non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot. . , i.e. the word w is obtain from w by changing each of its letters by its reverse. Of course this choice is strictly equivalent to presentation (1). But in a sense, when m > 0 the word w m does not give a "reduced" relation (some cancelations are possible in w m xw −m y −1 ) which is not the case for the word Ω m . Some more elementary properties of twist knots are discussed in the following remark. 3 6 7

Left-orderable fundamental groups and Dehn surgery on genus one 2–bridge knots

“…Then the Riley polynomial of K is given by φ K (s, t) = z 1,1 + (1 − t)z 1,2 . (See also [8].) Since s and t are limited to be positive real numbers in our setting, it is not obvious that there exist solutions for Riley's equation φ K (s, t) = 0.…”

On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot

“…The normalization of the above mentioned Reidemeister torsion is the cohomological Reidemeister torsion associated with the meridian used in [23]. We note that the twisted Reidemeister torsion in [4] is the twisted Reidemeister torsion associated with the longitude, and it can be changed to the twisted Reidemeister torsion associated with the meridian by [29,Théorème 4.1] as mentioned in [23].…”

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

“…To choose a homology class for H 1 we pick up a simple closed curve on ∂ S 3 \ K . The following definition is due to Porti [37,Définition 3.21] (see also [4])…”

“…For a representation ρ : π 1 S 3 \ K → SL(2; C), put Φ := Ad ρ ⊗α, where α : π 1 (S 3 \ K) → Z ∼ = H 1 S 3 \ K is the Abelianization sending x i → t for any i, where t is a generator of Z and we denote Φ(x i ) by t Ad ρ(xi) , noting that x i is sent to the generator of Z by α. Now we follow the technique used in [4]. We use the following theorem.…”