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The Bing double of a knot is defined by placing the Bing double of the unknot in the
In knot theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
, a field of mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Bing double of a knot is a link with two components which follow the pattern of the knot and "hook together". Bing doubles were introduced in by their namesake, the American mathematician R. H. Bing. The Bing double of a slice knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
Definition
A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the ...
is a slice link, though it is unknown whether the converse is true. The components of a Bing double bound disjoint Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link.
Such surfaces can be used to study the properties of the associated knot or link. For exampl ...
s.
The Bing double of a knot is defined by placing the Bing double of the unknot in the solid torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology.
A standard way to visual ...
surrounding it, as shown in the figure, and then twisting that solid torus into the shape of . This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the Bing link.
See also
*Slice knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
Definition
A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the ...
* Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link.
Such surfaces can be used to study the properties of the associated knot or link. For exampl ...
* Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include ...
* Whitehead double
References
Notes
Sources
*. *. *.Further reading
*. *. Knot theory Knot theory stubs {{knottheory-stub