1. 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
2. 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. However, 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
3. 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
4. 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.