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 potential for decorative purposes in Celtic-style ornamental knotwork. [3]
See also
- Wild arc
- Alexander horned sphere
- Eilenberg–Mazur swindle, a technique for analyzing connected sums using infinite sums of knots
References
- ^ Voitsekhovskii, M. I. (December 13, 2014) [1994], "Wild knot", Encyclopedia of Mathematics, EMS Press
- ^ 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
- ^ Browne, Cameron (December 2006), "Wild knots", Computers & Graphics, 30 (6): 1027–1032, doi: 10.1016/j.cag.2006.08.021
| Hyperbolic |
|
|---|---|
| Satellite | |
| Torus | |
| Invariants | |
| Notation and operations | |
| Other | |
 | This knot theory-related article is a stub. You can help Wikipedia by expanding it. |
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 potential for decorative purposes in Celtic-style ornamental knotwork. [3]
See also
- Wild arc
- Alexander horned sphere
- Eilenberg–Mazur swindle, a technique for analyzing connected sums using infinite sums of knots
References
- ^ Voitsekhovskii, M. I. (December 13, 2014) [1994], "Wild knot", Encyclopedia of Mathematics, EMS Press
- ^ 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
- ^ Browne, Cameron (December 2006), "Wild knots", Computers & Graphics, 30 (6): 1027–1032, doi: 10.1016/j.cag.2006.08.021
| Hyperbolic |
|
|---|---|
| Satellite | |
| Torus | |
| Invariants | |
| Notation and operations | |
| Other | |
|
| This knot theory-related article is a stub. You can help Wikipedia by expanding it. |