51. CMB 1998 (vol 41 pp. 140)
|Skein homology |
A new class of homology groups associated to a 3-manifold is defined. The theories measure the syzygies between skein relations in a skein module. We investigate some of the properties of the homology theory associated to the Kauffman bracket.
52. 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.
53. CMB 1997 (vol 40 pp. 370)
|Which $3$-manifolds embed in $\Triod \times I \times I$? |
We classify the compact $3$-manifolds whose boundary is a union of $2$-spheres, and which embed in $T \times I \times I$, where $T$ is a triod and $I$ the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any $3$-manifold in $T \times I \times I$ embeds in a punctured handlebody.
Categories:57N10, 57N35, 57Q35
54. CMB 1997 (vol 40 pp. 204)
|The $\eta$-invariants of cusped hyperbolic $3$-manifolds |
In this paper, we define the $\eta$-invariant for a cusped hyperbolic $3$-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.
Categories:57M50, 53C30, 58G25