Knots in the Solid Torus

Warning

It is different with the Knots in the Torus.

Solid Torus

Definition

Definition

A solid torus is a space homeomorphic to ${\mathbb{S}}^1\times {\mathbb{D}}^2$. I am going to denote it as $V$.

Definition

Framing of Solid Torus is a specific homeomorphism $h:{\mathbb{S}}^1\times {\mathbb{D}}^2\to V$ is called a framing of $V$.

Since it has a boundary $\partial V$ which is a ${\mathbb{T}}^2$. So, we can use the knowledge in Knots in the Torus to define the meridian and the longitude in the =solid torus=.¹

From here, I do not consider the inessential knot. Every things are [[knots in the torus# Knot types of ${\mathbb{S}}^1$ in ${\mathbb{T}}^2$|essential knots]] and simple closed curve in $\partial V$.

Meridian in the Solid Torus

Definition

A meridian $\mu \subset V$, is the curve which bounds the disc in the $V$². Also correspond $\mu$ to the $\left(0,1\right)$ in the universal cover of $\partial V$.

Equivalent Definitions

It has equivalent definitions

1. Homotopically, homologically trivial in $V$.
2. For some framing $h:{\mathbb{S}}^1\times {\mathbb{D}}^2\to V$, $\mu =h\left(1\times \partial {\mathbb{D}}^2\right)$.

Longitude in the Solid Torus

Definition

Definition

A longitude $l\subset V$, is any simple closed curve in $\partial V$ of the form $h\left({\mathbb{S}}^1\times 1\right)$, for some framing $h$ of $V$.

Equivalent Definitions

1. $l$ is a generator of $H_1\left(V\right)={\pi }_1\left(V\right)=\mathbb{Z}$.
2. $l$ intersects some meridian of $U$ transversally in a single point.

Meridian is Intrinsic and the Longitude is the Choice

Meridian

Any two meridians of $V$ are equivalent by an ambient isotopy of $V$ in $S^3$.

Also, if we can extend a homeomorphism $h:\partial V\to \partial V$ to $\tilde{h}:V\to V$ if and only if meridians go to the meridians.

Longitude

On the other hand, any two longitudes are equivalent by a homeomorphism of $V$; however, there are infinitely many ambient isotopy classes.

Preferred Framing

Using linking number, we can choose the how much revolutions we need.

Application

Satellite Knot

Footnotes

1. It is different with the torus case. ↩

2. do not confuse with ${\mathbb{S}}^3$. ↩