多面体环：T(3,3)和T(6,3)

Torus links: T(3,3) and T(6,3)

2023-08-26 00:17:07

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Summary

These torus links were constructed by Hillis Burns, Shannon Timoney, Hall Pritchard (students in Math 383D Knot Theory Spring 2023).

A torus knot or link is a curve which is embedded on a torus (the mathematical name for the surface of a doughnut). These curves can be distinguished by the number of times the curve winds around the long way about the torus and then number of ways the curve winds around the short way. The T(p,q) torus knot/link winds p times around the long way and q times around the short way.

When p and q have no common factors, for example T(2,11), the curve is a knot. When p and q have common factors, the curve is a link. For example, T(2,6) is a 2 component link an each component winds once around the long way and three times around the short way. Many examples can be found on the Knot Plot website: https://knotplot.com/knot-theory/torus_xing.html

The torus links here are the following:
• T(3,3) which has 3 components, each of which winds once around the torus the long way and once around the short way.
• The T(3,3) torus link on the torus.
• T(6,3) which has 3 components, each of which winds once around the torus the long way and twice around the short way.

More on the math and construction of torus knots and links can be found here: https://mathvis.academic.wlu.edu/2023/05/19/overview-of-torus-shapes-knots-and-links/

Further details on constructing the torus links in Cinema 4D can be found here: https://mathvis.academic.wlu.edu/2023/05/19/new-torus-link-improved-visualizations-and-cinema-4d-problems/