Topic
Solid torus
About: Solid torus is a research topic. Over the lifetime, 403 publications have been published within this topic receiving 7067 citations.
Papers published on a yearly basis
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Papers
TL;DR: In this article, it was shown that a 3-manifold can be constructed by r-Dehn surgery on a fc-component link, where the re-attached solid torus neighborhood is parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the reattached torus.
Abstract: In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that
756 citations
TL;DR: In this paper, the set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [2], [13] and [15] ).
538 citations
TL;DR: In this paper, it was shown that the fundamental group of the two-branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components.
Abstract: Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in $$\text{ Homeo}_+(S^1)$$
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259 citations
TL;DR: The Conway and Kauffman modules of a solid torus are calculated in this paper, where the authors show that they are the same as the ones of the torus of the Earth.
Abstract: The Conway and Kauffman modules of a solid torus are calculated.
234 citations
TL;DR: The main purpose of as mentioned in this paper is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1].
Abstract: The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above.
232 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 15 |
| 2020 | 17 |
| 2019 | 12 |
| 2018 | 18 |
| 2017 | 10 |