Tags » Low-dimensional Topology

Hyperbolic Knots


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

Low-Dimensional Topology

Products of Links, Satellite Construction

*Ref: Cromwell

Satellite Construction

We first define some terms for links.

Definition 1:

A link is called a spilt linkĀ if 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 .

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Low-Dimensional Topology

Links Diagram, Reidemeister Moves

*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

Low-Dimensional Topology

A little bit of Surgery and Loops

*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…

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Low-Dimensional Topology

Torus Knots

This 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

Low-Dimensional Topology

Basic Concepts of Knots and Links

*Reference: Cromwell – Knots and Links

Knots and Links

Definition 1:

A knot is 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.

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Low-Dimensional Topology

Some Weird Spaces

Alexander’s Horned Sphere

A topological whose unbounded complement is not simply connected.

Antoine’s Horned Sphere

A topological whose exterior and outer complement is not simply connected. 16 more words

Low-Dimensional Topology