Advertisement

Circle pair

A circle pair is a one dimensional shape constructed from a pair of two disconnected circles. This shape is found as the boundary of some higher dimensional shapes, such as the two-dimensional tube.

Knot Theory

In knot theory, embeddings of the circle pair into the Euclidean realm make up the 2-component links. These are distinct in that no link can be smoothly deformed into another without intersecting itself at some point

Unlink

The simplest link, the unlink, is the link made up of a circle pair where the components are completely disconnected and seperate.

The unlink can be seperated out into two distinct unknots.

Hopf Link

L2a1 Link.png
Hopf link
The Hopf link is the simplest nontrivial link, made up of a circle pair where the two circles are linked together exactly once but are otherwise undistorted.

The Hopf link can be seperated out into two distinct unknots.

Solomon's Knot

Solomon's knot is a link made up of a circle pair in which the circles are doubly linked.

Solomon's knot can be seperated out into two distinct unknots

Whitehead Link

The Whitehead link is a link made up of a circle pair, with one of the circles twisted into a figure-eight shape intertwined with a larger circle which goes through the two loops of the figure-eight.

The Whitehead link can be seperated out into two distinct unknots; the inner knot, twisted into a figure-eight shape, is only unable to be deformed into an unknot normally because it is linked with the outer knot which constrains it.

Community content is available under CC-BY-SA unless otherwise noted.