scholarly journals ALEXANDER POLYNOMIALS OF RIBBON LINKS

2011 ◽  
Vol 20 (02) ◽  
pp. 327-331
Author(s):  
JONATHAN A. HILLMAN
Keyword(s):  
Alexander Polynomial ◽  
Simple Argument ◽  
Alexander Polynomials

We give a simple argument to show that every polynomial f(t) ∈ ℤ[t] such that f(1) = 1 is the Alexander polynomial of some ribbon 2-knot whose group is a 1-relator group, and we extend this result to links.

Differences of Alexander polynomials for knots caused by a single crossing change, II

2017 ◽  
Vol 26 (14) ◽  
pp. 1750097
Author(s):  
Yasutaka Nakanishi
Keyword(s):  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Single Crossing ◽  
Previous Note

In the previous note, Okada and the author gave an approach to give a characterization of Alexander polynomials for knots which are transformed by a single crossing change into a given knot whose Alexander polynomial is monic. In this note, we give a characterization in the case of [Formula: see text], and show that the Gordian distance of [Formula: see text] and [Formula: see text] is two. We also give a characterization in the cases of more three knots.

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 A deformation of the Alexander polynomials of knots yielding lens spaces

2007 ◽  
Vol 75 (1) ◽  
pp. 75-89 ◽  
Author(s):  
Teruhisa Kadokami ◽  
Yuichi Yamada
Keyword(s):  
Lens Space ◽  
Alexander Polynomial ◽  
Lens Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reidemeister Torsion ◽  
Alexander Polynomials ◽  
Torus Knot ◽  
Necessary And Sufficient

For a knot K in a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold of p/q-surgery along K. We say that the manifold or the surgery is of lens type if Σ(K;p/q) has the same Reidemeister torsion as a lens space.We prove that, for Σ(K;p/q) to be of lens type, it is a necessary and sufficient condition that the Alexander polynomial ΔK(t) of K is equal to that of an (i, j)-torus knot T(i, j) modulo (tp – 1).We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion as L(p, q') then (1) (2) The multiple of ΣK(tk) over k ∈ (i) is ±tm modulo (tp – 1), where (i) is the subgroup in (Z/pZ)×/{±1} generated by i. Conversely, if a subgroup H of (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k ∈ H) is ±tm modulo (tp – 1), then H includes i or j.Here, i, j are the parameters of the torus knot above.

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 Explicit formulae for two-bridge knot polynomials

2005 ◽  
Vol 78 (2) ◽  
pp. 149-166 ◽  
Author(s):  
Shinji Fukuhara
Keyword(s):  
Normal Form ◽  
Open Problem ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Elementary Number ◽  
Conway Polynomial ◽  
Coprime Integers ◽  
Explicit Formulae ◽  
Knots And Links

AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS

2009 ◽  
Vol 18 (07) ◽  
pp. 973-984 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Prime Power ◽  
Alexander Polynomial ◽  
Branched Covers ◽  
Splitting Property ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Knot Concordance ◽  
Branched Cover ◽  
Linearly Independent

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

Alexander polynomials of two-bridge links

1984 ◽  
Vol 36 (1) ◽  
pp. 59-68 ◽  
Author(s):  
Taizo Kanenobu
Keyword(s):  
Necessary Conditions ◽  
Alexander Polynomial ◽  
Alexander Polynomials ◽  
Monotonicity Conditions

AbstractWe provide an algorithm for calculating the Alexander polynomial of a two-bridge link by putting every two-bridge link in a special type of Conway diagram. Using this algorithm, some necessary conditions for a polynomial to be the Alexander polynomial of a two-bridge link are given, in particular, certain alternating and monotonicity conditions on the coefficients, analogous to corresponding known properties of the reduced Alexander polynomial.

scholarly journals Burau Maps and Twisted Alexander Polynomials

2018 ◽  
Vol 61 (2) ◽  
pp. 479-497
Author(s):  
Anthony Conway
Keyword(s):  
Braid Group ◽  
Alexander Polynomial ◽  
Burau Representation ◽  
Alexander Polynomials

AbstractThe Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

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.

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