Wild knot
In themathematical theory of knots,a knot istameif it can be "thickened", that is, if there exists an extension to anembeddingof thesolid torusinto the3-sphere.A knot is tame if and only if it can be represented as afiniteclosed polygonal chain.Every closed curve containing awild arcis a wild knot.[1]Knots that are not tame are calledwildand can havepathologicalbehavior. In knot theory and3-manifoldtheory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
It has been conjectured that everywild knothas infinitely manyquadrisecants.[2]
As well as their mathematical study, wild knots have also been studied for their potential for decorative purposes inCeltic-styleornamental knotwork.[3]
See also
[edit]- Wild arc
- Alexander horned sphere
- Eilenberg–Mazur swindle,a technique for analyzingconnected sumsusing infinite sums of knots
References
[edit]- ^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,MR1265452,S2CID6103528
- ^Browne, Cameron (December 2006), "Wild knots",Computers & Graphics,30(6): 1027–1032,doi:10.1016/j.cag.2006.08.021
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