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1. CMB Online first
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 |
2. CMB 2014 (vol 57 pp. 431)
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 |
3. CMB 2013 (vol 57 pp. 526)
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 |
4. CMB 2010 (vol 54 pp. 147)
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 |
5. CMB 2006 (vol 49 pp. 55)
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 |