solid torus

A standard way to picture a solid torus is as a toroid, embedded in 3-space.

Each embedding should be an unknotted solid torus in the 3-sphere.

Then tie up the solid torus into a nontrivial knot.

Now find a compact unknotted solid torus T inside the sphere.

The inner ring and the five outer rings now form a six ring, 60-cell solid torus.

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary.

Since the disk is contractible, the solid torus has the homotopy type of .

Let N be a thickened neighborhood of K; so N is a solid torus.

Since is an unknotted solid torus, is a tubular neighbourhood of an unknot .

The closed complement of the solid torus inside S is another solid torus.

Given a 3-manifold with torus boundary components, we may glue in a solid torus by a homeomorphism (resp.

The leaf is a torus T bounding a solid torus with the Reeb foliation.

A satellite knot can be picturesquely described as follows: start by taking a nontrivial knot lying inside an unknotted solid torus .

The knot group of the unknot is an infinite cyclic group, and the knot complement is homeomorphic to a solid torus.

If a knot is a satellite knot with companion i.e.: there exists an embedding such that where is an unknotted solid torus, then .

(A solid torus is an ordinary three-dimensional doughnut, i.e. a filled-in torus, which is topologically a circle times a disk.)

In the 3-manifold case, a picturesque description of a lens space is that of a space resulting from gluing two solid torus together by a homeomorphism of their boundaries.

Let denote the manifold obtained from M by filling in the i-th boundary torus with a solid torus using the slope where each pair and are coprime integers.

In the mathematical theory of knots, a framed knot is the extension of a tame knot to an embedding of the solid torus D x S in S.

Here "nontrivial" means that the knot is not allowed to sit inside of a 3-ball in and is not allowed to be isotopic to the central core curve of the solid torus.

In mathematics, a solid torus is a topological space homeomorphic to , i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology.

Now take a second solid torus T inside T so that T and a tubular neighborhood of the meridian curve of T is a thickened Whitehead link.

The central core curve of the solid torus is sent to a knot , which is called the "companion knot" and is thought of as the planet around which the "satellite knot" orbits.

A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds.

We can specify a homeomorphism of the boundary of a solid torus to T by having the meridian curve of the solid torus map to a curve homotopic to .