scholarly journals TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250138 ◽  
Author(s):  
JIM HOSTE ◽  
PATRICK D. SHANAHAN
Keyword(s):  
Alexander Polynomial ◽  
Torus Knots ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Genus One

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

scholarly journals Adjoint twisted Alexander polynomials of genus one two-bridge knots

2016 ◽  
Vol 25 (11) ◽  
pp. 1650065 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Alexander Polynomial ◽  
Explicit Formulas ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Genus One

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

scholarly journals Twisted Alexander polynomials with the adjoint action for some classes of knots

2014 ◽  
Vol 23 (10) ◽  
pp. 1450051 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Knot Theory ◽  
Adjoint Action ◽  
Alexander Polynomial ◽  
Reidemeister Torsion ◽  
Torus Knots ◽  
Alexander Polynomials ◽  
Fibered Knots ◽  
Twisted Alexander Polynomial ◽  
The One ◽  
Better Than

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

Adjoint twisted Alexander polynomial of twisted Whitehead links

2018 ◽  
Vol 27 (04) ◽  
pp. 1850026
Author(s):  
Hoang-An Nguyen ◽  
Anh T. Tran
Keyword(s):  
Knot Theory ◽  
Three Dimensional ◽  
Adjoint Action ◽  
Alexander Polynomial ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Whitehead Link ◽  
Twisted Alexander Polynomial ◽  
Genus One

The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500 ]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

scholarly journals Metabelian SL(n, ) representations of knot groups IV: twisted Alexander polynomials

2013 ◽  
Vol 156 (1) ◽  
pp. 81-97 ◽  
Author(s):  
HANS U. BODEN ◽  
STEFAN FRIEDL
Keyword(s):  
Tensor Product ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

AbstractIn this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

scholarly journals TWISTED ALEXANDER POLYNOMIALS AND CHARACTER VARIETIES OF 2-BRIDGE KNOT GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250022 ◽  
Author(s):  
TAEHEE KIM ◽  
TAKAYUKI MORIFUJI
Keyword(s):  
Alexander Polynomial ◽  
Character Variety ◽  
Character Varieties ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

We study the twisted Alexander polynomial from the viewpoint of the SL (2, ℂ)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL (2, ℂ)-representations are all monic. In this paper, we show that for a 2-bridge knot there exists a curve component in the SL (2, ℂ)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g - 2.

scholarly journals TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350022 ◽  
Author(s):  
TAEHEE KIM ◽  
TAKAHIRO KITAYAMA ◽  
TAKAYUKI MORIFUJI
Keyword(s):  
Alexander Polynomial ◽  
Character Variety ◽  
Character Varieties ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, ℂ)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, ℂ)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

2018 ◽  
Vol 27 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Mikami Hirasawa ◽  
Kunio Murasugi
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial ◽  
Polynomial Formula ◽  
Integer Polynomial

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

scholarly journals Twisted Alexander polynomials of torus links

2020 ◽  
Vol 29 (04) ◽  
pp. 2050016
Author(s):  
Teruaki Kitano ◽  
Takayuki Morifuji ◽  
Anh T. Tran
Keyword(s):  
Explicit Formula ◽  
Alexander Polynomial ◽  
Constant Function ◽  
Character Variety ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Higher Dimensional ◽  
Twisted Alexander Polynomial ◽  
Torus Links

In this paper, we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the [Formula: see text]-character variety. We also discuss similar things for the higher-dimensional twisted Alexander polynomial and the Reidemeister torsion.

EVALUATIONS OF THE TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS AT ±1

2010 ◽  
Vol 19 (10) ◽  
pp. 1355-1400 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
KUNIO MURASUGI
Keyword(s):  
Algebraic Integer ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Twisted Alexander Polynomial

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s0 such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s0]) ⊂SL(2, ℂ). Let [Formula: see text] be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), [Formula: see text] and [Formula: see text], where [Formula: see text], μ ∈ ℤ[s0]. The number μ can be recursively evaluated.

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