Solid Torus Research Articles

Notes on Euclidean de Sitter space

We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a three-hemisphere. However, when de Sitter entropy is computed in a `stretched horizon' picture, then we argue that the correct Euclidean topology is a solid torus. The solid torus shrinks and degenerates into a three-hemisphere as one goes from the `stretched horizon' to the horizon, giving the Euclidean continuation of the causal diamond. We finally comment on the generalisation of these results to higher dimensions.

On the Kauffman skein modules

Let k be a subring of the field of rational functions in α, s which contains α ±1 ,s ±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S 1 × S 2 is freely generated by the empty link, i.e., .

Quantum gravity on dS3

We study quantum gravity on dS3 using the Chern–Simons formulation of three-dimensional gravity. We derive an exact expression for the partition function for quantum gravity on dS3 in a Euclidean path integral approach. We show that the topology of the space relevant for studying de Sitter entropy is a solid torus. The quantum fluctuations of de Sitter space are sectors of configurations of point masses taking a discrete set of values. The partition function gives the correct semiclassical entropy. The sub-leading correction to the entropy is logarithmic in horizon area, with a coefficient −1. We discuss this correction in detail, and show that the sub-leading correction to the entropy from the dS/CFT correspondence agrees with our result. A comparison with the corresponding results for the AdS3 BTZ black hole is also presented.

Contact topology and hydrodynamics II: solid tori

We prove the existence of periodic orbits for steady real-analytic Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We prove the Weinstein Conjecture on the solid torus via a combination of results due to Hofer et al and a careful analysis of tight contact structures on solid tori.

ON MARKOV'S THEOREM

Let χ be an oriented link type in the oriented 3-sphere S3 or ℝ3 = S3 - {∞}. A representative X ∈ χ is said to be a closed braid if there is an unknotted curve A ⊂ S3 - X (the axis) and a choice of fibration ℋ of the open solid torus S3 - A by meridian discs {Hθ : θ ∈ [0, 2 π]}, such that whenever X meets a fiber Hθ the intersection is transverse. Closed braid representations of χ are not unique, and Markov's well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov's theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.

SEIFERT UNIONS AND SPACES OF GRAPHS IN S3

Let Γ be a non-splittable in S3, not a tree. It is shown that if the space [Formula: see text] of Γ is a Seifert union of solid tori then M is homeomorphic to the space of a tree of Burde-Murasugi links.

Differentiable Family of Contact Structures on a Non-Compact (2n-1)-Manifold

We construct a differentiable family of non-contactomorphic contact structures on a non-compact (2n-1)-dimensional manifold. This generalizes a result of Eliashberg which shows the existence of non-contactomorphic open solid tori.

Singhof fillings of closed 3-manifolds

We give a complete classification of all closed, connected 3-manifolds which admit a Singhof filling with any number of solid tori.

An index for closed orbits in Beltrami fields

We consider the class of Beltrami fields (eigenfields of the curl operator) on three-dimensional Riemannian solid tori: such vector fields arise as steady incompressible inviscid fluids and plasmas. Using techniques from contact geometry, we construct an integer-valued index for detecting closed orbits in the flow which are topologically inessential (they have winding number zero with respect to the solid torus). The most important feature of the index is its computability: it can be rigorously determined from a C 1-approximation to the vector field on any meridional disc of the solid torus. As a result, we obtain a test for detecting the non-existence of a global cross-section to the 3D flow via purely 2D information.

Layer manufacture of jewel ring by volume efficient methodology

Rapid prototyping (RP) technology has been widely used in making prototypes for quite a long time. However, an in-depth study of methods used to fabricate jewellery products with RP is not available. In this paper, a new direct and adaptive slicing methodology is applied to three-dimensional solid jewel ring models to produce maximum inscribed three-dimensional stacked slab models. Jewel rings that are symmetrical in one or more principal planes XY, YZ and ZX are considered. The maximum inscribed slab model will minimize material wastage in post-treatment processes and preserve the symmetric property of the original jewel ring model. A solid torus is used as a simplified representation of the jewel ring in the description of the methodology. A case study based on the methodology is presented.

On the Homflypt skein module of $S^1 \times S^2$

Let k be a subring of the field of rational functions in x, v,s which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$ . If M is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$ ; (2) L with a positiv e twist $=(xv^{-1})L$ ; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of $S(S^1\times D^2)$ given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for $S(S^1\times D^2)$ based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements $s^{2n}-1$ , for n a nonzero integer, and the elements $s^{2m}-v^{2}$ , for any integer m, are invertible ink, then $S(S^{1} \times S^2)=k$ -torsion module $\oplus k$ . Here the free part is generated by the empty link $\phi$ . In addition, if the elements $s^{2m}-v^{4}$ , for m an integer, are invertible in k, then $S(S^{1} \times S^2)$ has no torsion. We also obtain some results for more general k.

Plane curves, wavefronts and Legendrian knots

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Goryunov V. V. 2001Plane curves, wavefronts and Legendrian knotsPhil. Trans. R. Soc. A.3591497–1510http://doi.org/10.1098/rsta.2001.0837SectionRestricted accessPlane curves, wavefronts and Legendrian knots V. V. Goryunov V. V. Goryunov Department of Mathematical Sciences, Divison of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK () Google Scholar Find this author on PubMed Search for more papers by this author V. V. Goryunov V. V. Goryunov Department of Mathematical Sciences, Divison of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 July 2001https://doi.org/10.1098/rsta.2001.0837AbstractWe survey some of the recent results on Legendrian knots and links in the standard contact 3–space and solid torus. These include the description of finite–order invariants and estimates of the self–linking number coming from the classical polynomial link invariants. We also describe the combinatorial invariant introduced by Chekanov and Pushkar, which allowed them to prove Arnold's conjecture on the necessity of four–cusp curves in generic eversions of a circular front in the plane. Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by SAYGILI K (2012) TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORY, International Journal of Modern Physics A, 10.1142/S0217751X08039840, 23:13, (2015-2035), Online publication date: 20-May-2008. Thornby J, MacKay R and Williams M (2015) Mathematical principles for the design of isostatic mount systems for dynamic structures, IMA Journal of Applied Mathematics, 10.1093/imamat/hxv020, (hxv020) This Issue15 July 2001Volume 359Issue 1784Discussion Meeting Issue ‘Topological methods in the physical sciences’ organized by V. I. Arnold, J. W. Bruce, H. K. Moffatt and R. B. Pelz Article InformationDOI:https://doi.org/10.1098/rsta.2001.0837Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/07/2001Published in print15/07/2001 License: Citations and impact KeywordsLegendrian linksChekanov–Pushkar invariantpolynomial knot invariantsBennequin numberarnold–s conjecturefinite–type invariants

Cyclotomic Birman–Murakami–Wenzl algebras

We introduce an extension of the Birman–Murakami–Wenzl Algebra that is related to the braid group of the cylinder and show it to be semi-simple in the generic case. This algebra supports a faithful Markov trace that gives rise to link invariants in the solid torus.

MINIMISING THE BOUNDARIES OF PUNCTURED PROJECTIVE PLANES IN S3

This paper concerns 3-manifolds X obtained by Dehn surgery on a knot in S 3, in particular those which contain embedded projective planes. Either, they are homeomorphic to the 3-real projeclive space ℝ P 3, or they are reducible. Let p be the number of intersections of a projective plane in X with the core of the solid torus added during surgery. We prove here that either X is reducible or p is bigger than or equal to five. Consequently, if X is homeomorphic to ℝ P 3 then all its projective planes are pierced at least in five points by the core of the surgery. This result is considered as a step towards showing that ℝ P 3 cannot be obtained by a Dehn surgery along a knot in S 3.

Actions of tensor categories, cylinder braids and their Kauffman polynomial

Categorial actions of braided tensor categories are defined and shown to be the right framework for a discussion of the categorial structure related to the group of braids in the cylinder. A Kauffman polynomial of links in the solid torus is constructed. It is related to a Birman–Wenzl algebra.

Heegaard splittings of the pair of the solid torus and the core loop

We show that any Heegaard splitting of the pair of the solid torus (@D2?S1) and its core loop (an interior point ?S1) is standard, using the notion of Heegaard splittings of pairs of 3-manifolds and properly imbedded graphs, which is defined in this paper.

Linked-solenoid mappings and the non-transversality locus invariant

We will study in this paper diffeomorphisms f : S3 → S3 with two invariant linked solenoids Σ+, Σ−, one attracting and one repelling. We will call such mappings linked solenoid mappings. Such mappings arise when studying Henon mappings in C2, and have been studied in [HO1], [HPV], [BS], and [Bu]. More precisely, a linked solenoid mapping is one for which the 3-sphere S3 can be cut into two linked unknotted solid tori T+, T−, such that f : T+ → T+ and f−1 : T−1 → T−1 are conjugate to the standard maps (see below) giving rise to solenoids, as first studied in [vD] and [V]. These maps are structurally stable, and hence our linked solenoid maps will be structurally stable on their non-wandering sets. But they are not structurally stable. In Section 3 of this paper we will define a conjugacy invariant ntl(f ) ⊂ (R/2πZ)2 of linked solenoid mappings mapping. We will compute ntl(f ) for some particular linked solenoid maps in Section 4, we will show it can take on an infinite-dimensional set of values in Section 5, and in Section 6 we will show that in an open set of solenoidal mappings it is a complete invariant and classifies these mappings up to topological conjugacy. There are many things about this invariant which we don’t know; we present some of these in Section 7. We thank Thierry Bousch, whose criticism of [HO1] prompted part of this paper.

How a strongly irreducible Heegaard splitting intersects a handlebody

In [Topology Appl. 90 (1998) 135] Scharleman showed that a strongly irreducible Heegaard splitting surface Q of a 3 -manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M .

Persistence of Heegaard structures under Dehn filling

It is well known that a Heegaard surface may destabilize after Dehn filling, reducing the genus by one or more. This phenomenon is classified according to whether or not the core of the attached solid torus is isotopic into the destabilized surface. When it is, the destabilized surface will be a Heegaard surface for infinitely many fillings, arranged along a destabilization line in the Dehn surgery space. Here we demonstrate that a destabilization line corresponds to a slope bounding an essential surface. Such slopes are known to be finite in number and therefore so is the number of destabilization lines. We apply this result to study Heegaard genus. In particular we prove, using purely topological techniques, that if X is any a-cylindrical manifold, then there are an infinite number of Dehn fillings on X which produce a manifold of the same genus as X .

THE THIRD POWER OF THE SMALE HORSESHOE INDUCES ALL LINK TYPES

Let φ:D2→D2 be an orientation preserving homeomorphism of the disk into itself, and Φ= {φt}0≤t≤1 an isotopy with φ0=idD2 and φ1=φ. Then for a finite union of periodic orbits P of φ, the set [Formula: see text] is a link in D2×S1. We say that φ induces all link types (for Φ) if there exists a homeomorphism h of D2×S1 into a standardly embedded solid torus in the 3-sphere S3 such that any link L in S3 can be realized by a finite union of periodic orbits PL of φ so that L and [Formula: see text] are equivalent. We will show that the Smale horseshoe and its second power do not induce all link types, but its third power does induce all link types.