*Ref:Cromwell

A knot is called a **hyperbolic knot** if its complement is a hyperbolic space. Recall that a surface is called **hyperbolic** if it admits a Riemannian metric such that the Gaussian curvature is identical to . 375 more words

*Ref:Cromwell

A knot is called a **hyperbolic knot** if its complement is a hyperbolic space. Recall that a surface is called **hyperbolic** if it admits a Riemannian metric such that the Gaussian curvature is identical to . 375 more words

*Ref: Cromwell

We first define some terms for links.

**Definition 1:**

1,239 more wordsA link is called a

spilt linkĀif there exists embedded in such that the sphere separate into two connected component and such that one component of is in , the other one is in .

*Ref : Cromwell

We should like to deal with links in a 2-dimensional sense. A linking diagram is a graph that contains the information of crossings of a link, and we can recover the original link by just looking at the 2-dimensional diagram. 1,457 more words

*Reference: Cromwell

**Definition 1:**

A connected component of the boundary of a manifold is called a

boundary component.

The boundary components are the black loops…

630 more wordsThis are the knots that live on the torus surface.

A torus surface looks like

We can generate a torus by taking a circle on the -plane of radius r with center on -axis at distance from the origin, then rotate it around -axis. 342 more words

*Reference: Cromwell – **Knots and Links**

**Definition 1:**

A

knotis a subset of points homeomorphic to a circle .

Notice that the definition does not specify smoothness, so a knot can be represented as polygon (piecewise smooth) or just a smooth curve.

1,259 more words