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Search: All articles in the CJM digital archive with keyword knot

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1. CJM 2015 (vol 68 pp. 3)

Boden, Hans Ulysses; Curtis, Cynthia L
The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$-polynomial
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\widehat{A}$-polynomial of $K$. We prove a product formula for the $\widehat{A}$-polynomial of the connected sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity of ${SL(2, {\mathbb C})}$ Casson knot invariant under connected sum for a large class of knots in $S^3$. We also present an example of a nontrivial knot $K$ in $S^3$ with trivial $\widehat{A}$-polynomial and trivial ${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants detect the unknot.

Keywords:Knots, 3-manifolds, character variety, Casson invariant, $A$-polynomial
Categories:57M27, 57M25, 57M05

2. CJM 2011 (vol 64 pp. 102)

Ishii, Atsushi; Iwakiri, Masahide
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

3. CJM 2008 (vol 60 pp. 164)

Lee, Sangyop; Teragaito, Masakazu
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
Categories:57M50, 57N10

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