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 ${\displaystyle S^{1}\times D^{2}}$ into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.

A wild knot

Knots that are not tame are called wild and can have pathological behavior. Every closed curve containing a wild arc is a wild knot.^[1] 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