Wild knot

Wild node example

Wild knot is a pathological enclosure of a circle in space.

Wild knots can be found in some Celtic patterns.

Content

Definition

A knot is called tame if it can be “thickened,” that is, if it extends to a solid torus S ¹ × D ² that can be embedded in a 3-sphere . In knot theory and in the theory of , the word “manual” is often omitted.

Non-manual nodes are called wild and may have pathological behavior.

Examples

Doug Fox - Artin

Wild nodes are those containing the so-called arcs, some simple arcs obtained by wild embedding in $E^{3}$ ${\ displaystyle E ^ {3}}$ . For example, for an arc $L_{1}$ ${\ displaystyle L_ {1}}$ fundamental group $p_{1}$ ${\ displaystyle p_ {1}}$ ( $E^{3}L$ ${\ displaystyle E ^ {3} L}$ ) is nontrivial, for an arc $L_{2}$ ${\ displaystyle L_ {2}}$ Group ${\pi }_{1}(E^{3}/L)$ ${\ displaystyle {\ pi} _ {1} (E ^ {3} / L)}$ trivial but itself $E^{3}/L$ ${\ displaystyle E ^ {3} / L}$ not homeomorphic to the complement in $E^{3}$ ${\ displaystyle E ^ {3}}$ to the point ^[1] .

The figure shows a wild node with one wild (pathological) point. It is easy to build a wild knot containing several pathological points, an infinite number of such points, and even an uncountable set of pathological points. Sosinsky’s book ^[2] shows the construction of a wild node, the pathological points of which form a Cantor set . It is possible to imagine a wild knot containing a more complex set - the necklace of Antoine ^[2] .

Property

A node is tame if and only if it can be represented as a finite polyline .
Smooth knots are manual.

Variations and generalizations

Nontrivial wild knots also appear in the spheres of higher dimensions. For example, by the theorem on a double superstructure , a double superstructure over the Poincare sphere is homeomorphic to the standard sphere $\mathbb {S} ^{5}$ ${\ displaystyle \ mathbb {S} ^ {5}}$ . In this case, the equator of the double superstructure forms $\mathbb {S} ^{5}$ ${\ displaystyle \ mathbb {S} ^ {5}}$ wild knot and its complement has a nontrivial fundamental group .

Notes

↑ Wojciechowski M.I. Wild knot // Mathematical Encyclopedia / Ch. ed. I.M. Vinogradov. - M .: Soviet Encyclopedia, 1979. - T. 2. - S. [69] (stb. 137-138).
↑ ¹ ² Sosinsky, 2005 , p. 22.

Literature

LH Kauffman. An invariant of regular isotopy // Transactions of the American Mathematical Society. - American Mathematical Society, 1990. - Vol. 318, No. 2 .
A. B. Sosinsky. Nodes Chronology of a mathematical theory. - Moscow: ICSTMO, 2005. - ISBN 5-94057-220-0 .

[1] Wojciechowski M.I. Wild knot // Mathematical Encyclopedia / Ch. ed. I.M. Vinogradov. - M .: Soviet Encyclopedia, 1979. - T. 2. - S. [69] (stb. 137-138).

[_b91d89c75f7982e9-2] ¹ ² Sosinsky, 2005 , p. 22.