1. CMB Online first
||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
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
Keywords:twisted Alexander polynomial, virtual fibering theorem, unlink detection
2. 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
Keywords:finite quandles, finite biquandles, link invariants
3. CMB 2009 (vol 52 pp. 257)
||Essential Surfaces in Graph Link Exteriors |
An irreducible graph manifold $M$ contains an essential torus if
it is not a special Seifert manifold.
Whether $M$ contains a closed essential surface of
negative Euler characteristic or not
depends on the difference of Seifert fibrations from the two sides
of a torus system which splits $M$ into Seifert manifolds.
it is not easy to characterize geometrically the class of
irreducible graph manifolds which contain such surfaces.
This article studies this problem in the case of graph link exteriors.
Keywords:Graph link, Graph manifold, Seifert manifold, Essential surface
4. CMB 1999 (vol 42 pp. 190)
||Topological Quantum Field Theory and Strong Shift Equivalence |
Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$-dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The Turaev-Viro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the Silver-Williams
invariant, by using the TQFT associated to $G$ in its unmodified form.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence
Categories:57R99, 57M99, 54H20
5. CMB 1997 (vol 40 pp. 309)
||On the homology of finite abelian coverings of links |
Let $A$ be a finite abelian group and $M$ be a
branched cover of an homology $3$-sphere, branched over a link $L$,
with covering group $A$. We show that $H_1(M;Z[1/|A|])$ is determined
as a $Z[1/|A|][A]$-module by the Alexander ideals of $L$ and certain
ideal class invariants.
Keywords:Alexander ideal, branched covering, Dedekind domain,, knot, link.