A wild knot

In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. Every closed curve containing a wild arc is a wild knot.[1] Knots that are not tame are called wild and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.

It has been conjectured that every wild knot has infinitely many quadrisecants.[2]

As well as their mathematical study, wild knots have also been studied for their decorative purposes in Celtic-style ornamental knotwork.[3]

See also

References

  1. Voitsekhovskii, M. I. (December 13, 2014) [1994], "Wild knot", Encyclopedia of Mathematics, EMS Press
  2. Kuperberg, Greg (1994), "Quadrisecants of knots and links", Journal of Knot Theory and Its Ramifications, 3: 41–50, arXiv:math/9712205, doi:10.1142/S021821659400006X, MR 1265452, S2CID 6103528
  3. Browne, Cameron (December 2006), "Wild knots", Computers & Graphics, 30 (6): 1027–1032, doi:10.1016/j.cag.2006.08.021


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.