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Search: MSC category 57M27 ( Invariants of knots and 3-manifolds )

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1. CMB Online first

Tran, Anh T.; Yamaguchi, Yoshikazu
The asymptotics of the higher dimensional Reidemeister torsion for exceptional surgeries along twist knots
We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\operatorname{SL}_2(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly.

Keywords:Reidemeister torsion, graph manifold, asymptotic behavior, exceptional surgery
Categories:57M27, 57M50

2. CMB Online first

Friedl, Stefan; Vidussi, Stefano
Twisted Alexander invariants detect trivial links
It follows from earlier work of Silver--Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split and whether it is totally split.

Keywords:twisted Alexander polynomial, virtual fibering theorem, unlink detection

3. CMB 2016 (vol 59 pp. 472)

Clay, Adam; Desmarais, Colin; Naylor, Patrick
Testing Bi-orderability of Knot Groups
We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more.

Keywords:knots, fundamental groups, orderable groups
Categories:57M25, 57M27, 06F15

4. CMB 2015 (vol 59 pp. 159)

MacColl, Joseph
Rotors in Khovanov Homology
Anstee, Przytycki, and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants which can be distinguished by Khovanov homology, but not by the Jones polynomial.

Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomial
Categories:57M27, 57M25

5. CMB 2015 (vol 59 pp. 182)

Naylor, Geoff; Rolfsen, Dale
Generalized Torsion in Knot Groups
In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot $5_2$ and algebraic knots in the sense of Milnor.

Keywords:knot group, generalized torsion, ordered group
Categories:57M27, 32S55, 29F60

6. CMB 2014 (vol 57 pp. 431)

Tagami, Keiji
The Rasmussen Invariant, Four-genus and Three-genus of an Almost Positive Knot Are Equal
An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, $4$-genus and $3$-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$-genus one.

Keywords:almost positive knot, four-genus, Rasmussen invariant
Categories:57M27, 57M25

7. CMB 2013 (vol 57 pp. 526)

Heil, Wolfgang; Wang, Dongxu
On $3$-manifolds with Torus or Klein Bottle Category Two
A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this latter property are called $\mathcal{G}_K$-contractible. We obtain a list of the closed $3$-manifolds that can be covered by two open $\mathcal{G}_K$-contractible subsets. This is applied to obtain a list of the possible closed prime $3$-manifolds that can be covered by two open $K$-contractible subsets.

Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets
Categories:57N10, 55M30, 57M27, 57N16

8. CMB 2010 (vol 54 pp. 147)

Nelson, Sam
Generalized Quandle Polynomials
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants that further generalize the quandle counting invariant.

Keywords:finite quandles, finite biquandles, link invariants
Categories:57M27, 76D99

9. CMB 2006 (vol 49 pp. 55)

Dubois, Jérôme
Non Abelian Twisted Reidemeister Torsion for Fibered Knots
In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space, $\SU$, $\SL$, Adjoint representation, Monodromy
Categories:57Q10, 57M27, 57M25

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