Alexander Polynomial Of Knot Research Articles

A Fox–Milnor theorem for the Alexander polynomial of knotted 2-spheres in $S^4$

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon 2-knot in $S^4$ is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot that implies the factorization of the Alexander polynomial.

Knot adjacency from a surgical view of Alexander invariants

For two knots [Formula: see text] and [Formula: see text], [Formula: see text] is said to be [Formula: see text]-adjacent to [Formula: see text], if [Formula: see text] admits a knot diagram containing [Formula: see text] crossings such that crossing changes at any non-empty subset of them yield a knot diagram of [Formula: see text]. Using a surgical view of Alexander invariants, we will characterize the Alexander polynomials of knots which are [Formula: see text]-adjacent to the trivial knot.

A categorification of the Alexander polynomial in embedded contact homology

Given a transverse knot $K$ in a three dimensional contact manifold $(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,\alpha)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homologies in the plus and, respectively, hat versions.

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let \mathbb F be a field and let G \subset \mathbb F \setminus \{0\} be a multiplicative subgroup. We consider the category \mathsf {Cob}_G of 3 -dimensional cobordisms equipped with a representation of their fundamental group in G , and the category \mathsf {Vect}_{\mathbb F, \pm G} of \mathbb F -linear maps defined up to multiplication by an element of \pm G . Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from \mathsf {Cob}_G to \mathsf {Vect}_{\mathbb F, \pm G} . In particular, when the group G is free abelian and \mathbb F is the field of fractions of the group ring \mathbb Z [G] , we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for 3 -manifolds with boundary; when G is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup G\subset \mathbb F \setminus \{0\} . We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.

On stability of Alexander polynomials of knots and links (survey)

On stability of Alexander polynomials of knots and links (survey)

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

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.

META-MONOIDS, META-BICROSSED PRODUCTS, AND THE ALEXANDER POLYNOMIAL

We introduce a new invariant of tangles along with an algebraic framework in which it is to be interpreted. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it is the most meaningful. These are lecture notes for talks given by the first author, written and completed by the second. The talks, with handouts and videos, are available at http://www.math.toronto.edu/drorbn/Talks/Regina-1206/ . See also further comments at http://www.math.toronto.edu/drorbn/Talks/Caen-1206/#June8 .

Alexander Polynomials of Knots Which Are Transformed into the Trefoil Knot by a Single Crossing Change

By the works of Kondo and Sakai, it is known that Alexander polynomi- als of knots which are transformed into the trivial knot by a single crossing change are characterized. In this note, we will characterize Alexander polynomials of knots which are transformed into the trefoil knot (and into the gure-eight knot) by a single crossing change.

The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.

Zeros of the Alexander polynomial of knot

The leading coefficient of the Alexander polynomial of a knot is the most informative element in this invariant, and the growth of orders of the first homology of cyclic branched covering spaces is also a familiar subject. Accordingly, there are a lot of investigations into each subject. However, there is no study which deal with the both subjects in a same context. In this paper, we show that the two subjects are closely related in p-adic number theory and dynamical systems.

Nontrivial Alexander polynomials of knots and links

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguishing knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore we prove that these invariants decide if a genus one knot is fibered. Finally we also show that these invariants distinguish all mutants with up to 12 crossings.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

A deformation of the Alexander polynomials of knots yielding lens spaces

For a knotKin a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold ofp/q-surgery alongK. We say that the manifold or the surgery isof lens typeif 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 polynomialΔK(t)ofKis equal to that of an (i, j)-torus knotT(i, j)modulo (tp– 1).We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion asL(p, q')then (1)(2) The multiple of ΣK(tk) overk∈ (i) is ±tmmodulo (tp– 1), where (i) is the subgroup in (Z/pZ)×/{±1} generated byi. Conversely, if a subgroupHof (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k∈H) is ±tmmodulo (tp– 1), thenHincludesiorj.Here,i, jare the parameters of the torus knot above.

THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3are extended to the case of (1,1)-knots in lens spaces.

Yang–Baxter Operators Arising from Algebra Structures and the Alexander Polynomial of Knots #

ABSTRACT In this paper, we consider the problem of constructing knot invariants from Yang–Baxter operators associated to algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to derive knot invariants from these enhanced operators, as modified by Murakami, invariably produces the Alexander polynomial of knots.

On Milnor moves and Alexander polynomials of knots

On Milnor moves and Alexander polynomials of knots

A computer program to calculate Alexander polynomial from Braids presentation of the given knot

A central quantity for the calculation of Alexander polynomial of knots is to use Braids presentations of the given knots. For this purpose, we improved a computer program which is writting in Delphi programming language. The program calculates Alexander polynomials of the given knot using free derivative that is obtained from Braids presentation of the given knot.

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.

Representations of the singular braid monoid and group invariants of singular knots

It is well known that the Artin representation of the braid group may be used to calculate the fundamental group of knot complements while the Burau representation can be used to calculate the Alexander polynomial of knots. In this note we will study extensions of the Artin representation and the Burau representation to the singular braid monoid and the relation between them which are induced by Fox' free calculus. Closing singular braids, we obtain singular knots as they appear in the theory of Vassiliev invariants. Thus the extensions of the Artin representation and the Burau presentation give rise to invariants of singular knots. Here, we will focus on the invariants coming from the extended Artin representations. We obtain an infinite family of group invariants, all of them in relation with the fundamental group of the knot complement.