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

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1. CJM 2016 (vol 69 pp. 1201)

Abe, Tetsuya; Tagami, Keiji
Characterization of Positive Links and the $s$-invariant for Links
We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen and Beliakova-Wehrli's $s$-invariant. We also study almost positive links, in particular, determine the $s$-invariants of almost positive links. This result suggests that all almost positive links might be strongly quasipositive. On the other hand, it implies that almost positive links are never homogeneous links.

Keywords:knot, $s$-invariant, positive link, almost positive link
Categories:57M25, 57M27

2. 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

3. 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

4. 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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