Solid Torus Research Articles

Finding disjoint surfaces in 3-manifolds

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no two-spheres. We investigate the existence of two properly embedded disjoint surfaces \(S_{1}\) and \(S_{2}\) such that \(M - (S_{1} \cup S_{2})\) is connected. We show that there exist two such surfaces if and only if M is neither a \(\mathbb Z _{2}\) homology solid torus nor a \(\mathbb Z _{2}\) homology cobordism between two tori. In particular, the exterior of a link with at least three components always contains two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.

On singular foliations on the solid torus

We study smooth foliations on the solid torus S1×D2 having S1×{0} and S1×∂D2 as the only compact leaves and S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.

ON THE S1× S2HOMFLY-PT INVARIANT AND LEGENDRIAN LINKS

In [On the HOMFLY-PT skein module of S1× S2, Math. Z. 237(4) (2001) 769–814], Gilmer and Zhong established the existence of an invariant for links in S1× S2which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in S1× S2and shows that the invariant is in fact a Laurent polynomial in a and z = s – s-1. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston–Bennequin number to Legendrian links in S1× S2with its tight contact structure.

Jones's polynomial for links in the handlebody

In [3] Hoste and Przytycki defined a two variable polynomial invariant for 1-trivial dichromatic links by a method similar to that of Kauffman in defining the Jones polynomial. In this paper we view the invariant of Hoste and Przytycki as an invariant for knots and links in the solid torus, and we give a state sum formula for this invariant. Then we define Jones’s polynomial for links in the handlebody. In particular, we give formulas for the Jones’s polynomial for knots and links in the handlebody with two handles. Moreover, we give a state sum formula for this invariant.

On Krebesʼs tangle

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K⊆S3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 tangle G in a knot is given by torsion in the homology of branched covers of S branched over G. We examine a particular example A of a genus-1 tangle, given by Krebes, and consider its two double-branched covers. Using this homological obstruction, we show that any closure of A obtained via an arc which passes through the hole of S an odd number of times must have determinant divisible by three. A resulting corollary is that if A embeds in the unknot, then the arc which completes A to the unknot must pass through the hole of S an even number of times.

Dehn surgery, rational open books and knot Floer homology

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgeries are discussed.

A SEQUENCE OF NEW BRIDGE INDICES FOR LINKS EACH OF WHICH HAS A TRIVIAL KNOT COMPONENT

Let L = K1 ∪ K2 be a 2-component link in S3 such that K1 is a trivial knot. In this paper, we introduce for each n(∈ ℕ) a new bridge index bK1=n([L]) of L called a constrained bridge index with respect to n-bridge K1. Roughly speaking, bK1=n([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K1 are n. We give exact values bK1=n([L]) (n = 1, 2, …) for particular [Formula: see text] constructed as follows: There exist unknotted solid tori V1 ⊂ V2 ⊂ ⋯ ⊂ Vm in S3, where the core of each Vj(j = 1, 2, …, m - 1) is a (1, αj)-torus knot (αj ∈ ℤ) in Vj+1, such that K1 is the core of V1, and [Formula: see text] is the core of the exterior of Vm.

Metrics with prescribed Ricci curvature near the boundary of a manifold

Suppose \(M\) is a manifold with boundary. Choose a point \(o\in \partial M\). We investigate the prescribed Ricci curvature equation \(\mathop {\mathrm{Ric}}\nolimits (G)=T\) in a neighborhood of \(o\) under natural boundary conditions. The unknown \(G\) here is a Riemannian metric. The letter \(T\) on the right-hand side denotes a (0,2)-tensor. Our main theorems address the questions of the existence and the uniqueness of solutions. We explain, among other things, how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus \(\mathcal{T }\). The paper concludes with a brief discussion of the Einstein equation on \(\mathcal{T }\).

Dehn surgery on knots of wrapping number 2

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embedding $V$ in $S^3$ gives infinitely many knots $K_n$ with a slope $r_n$ corresponding to a slope $r$ of $K$ in $V$. If $r$ surgery on $K$ in $V$ is toroidal then either all but at most three $K_n(r_n)$ are toroidal, or they are all reducible or small Seifert fibered with two common singular fiber indices. These will be used to classify exceptional surgeries on wrapped Montesinos knots in solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.

Taylor series coefficients of the HP-polynomial as an invariant for links in the solid torus

We show that the coe-cients of a reformulation of a Taylor series expansion of the Hoste and Pryzyticki polynomial are Vassiliev invariants. We also show that many other reformulations of the Taylor series expansion have coe-cients that are Vassiliev invariants. We charecterize the flrst two coe-cients b 1(t) and b L (t) for one of these expan- sions. Moreover, the second coe-cient b L (t); which is a type-one Vassiliev invariant, is given two explicit computational formulas, which are easy to calculate. b L (t) is used to give a lower bound for the crossing number of a knot of zero

KNOTS IN THE SOLID TORUS UP TO 6 CROSSINGS

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

Legendrian and transverse cables of positive torus knots

In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" tori, which allow us to completely classify solid tori representing positive torus knots.

Generalized normal rulings and invariants of Legendrian solid torus links

For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.

THE MODULE OF TWO-CHORD DIAGRAMS FOR ORIENTED KNOTS OF ZERO WINDING NUMBER IN THE SOLID TORUS

In this paper we study the ℤ-module A2of two-chord diagrams for knots with zero winding number in the solid torus KST0, which is needed in studying the type-two invariants for knots in KST0. We show that this module (or abelian group), which is given as a presentation with infinite number of generators and an infinite number of relations, is a free infinitely generated module. Moreover, we show that this module is isomorphic to the direct sum of three free modules that are easier to understand.

THE GEOMETRY OF THE HANDLEBODY GROUPS I: DISTORTION

We show that the mapping class group of a handlebody V of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface ∂V. The same holds true for solid tori V with at least two marked points or spots.

Meridional number of a link and skein modules of the solid torus

We establish some relationships between the meridional number, an invariant of links in the solid torus, and the expression of these links in various skein modules of the solid torus. As an application, we calculate the meridional number of a family of links, the Bing links.

Periodic perturbation of quadratic systems with two infinite heteroclinic cycles

We study periodic perturbations of planar quadratic vector fieldshaving infinite heteroclinic cycles, consisting of an invariantstraight line joining two saddle points at infinity and an arc oforbit also at infinity. The global study concerning the infinity ofthe perturbed system is performed by means of the Poincarécompactification in polar coordinates, from which we obtain a systemdefined on a set equivalent to a solid torus in $\mathbb{R}^3$, whoseboundary plays the role of the infinity. It is shown that forcertain type of periodic perturbation, there exist twodifferentiable curves in the parameter space for which the perturbedsystem presents heteroclinic tangencies and transversalintersections between the stable and unstable manifolds of twonormally hyperbolic lines of singularities at infinity. Thetransversality of the manifolds is proved using the Melnikov methodand implies, via the Birkhoff-Smale Theorem, in a complex dynamicalbehavior of the perturbed system solutions in a finite part of thephase space. Numerical simulations are performed for a particularexample in order to illustrate this behavior, which could be called'the chaos arising from infinity', because it depends on theglobal structure of the quadratic system, including the points atinfinity.

Sources and sinks for Axiom A flows on closed 3-manifolds

We prove that an Axiom A vector field on an orientable closed 3-manifold not homeomorphic to S 3 for which every transverse torus bounds a solid torus either is transitive or has a sink or a source. This result is false without these hypotheses.

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function $f$ and study the simplest nontrivial case $f(x)=x^2$, obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.

The converse of the Solid Torus Theorem

Given positive integers p and q, a ( p , q ) - solid torus is a manifold diffeomorphic to D p + 1 × S q while a ( p , q ) - torus in a closed manifold M is the image of a differentiably embedding S p × S q → M . We prove that if n = p + q + 1 with p = q = 1 or p ≠ q , then M is homeomorphic to S n whenever every ( p , q ) -torus bounds a ( p , q ) -solid torus. We also prove for p = q that every closed n-manifold for which every ( p , p ) -torus bounds an irreducible manifold is irreducible. Consequently, every closed 3-manifold for which every torus bounds an irreducible manifold is irreducible.