1. CJM 2011 (vol 64 pp. 102)
|Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies|
We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.
Keywords:quandle cocycle invariant, knotted handlebody, spatial graph
Categories:57M27, 57M15, 57M25
2. CJM 2008 (vol 60 pp. 164)
|Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance|
|Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance |
For a hyperbolic $3$-manifold $M$ with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic $3$-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes. Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$ is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance $3$. We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.
Keywords:Dehn filling, annular filling, toroidal filling, knot