Bing double

In knot theory, a field of mathematics, 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 Bing (1952) by their namesake, the American mathematician R. H. Bing.^[1] The Bing double of a slice knot is a slice link, though it is unknown whether the converse is true.^[2] The components of a Bing double bound disjoint Seifert surfaces.^[2]

A solid torus encasing the Bing double of the unknot.

The unknot (left) and its Bing double (right).

The Bing double of a knot K is defined by placing the Bing double of the unknot in the solid torus surrounding it, as shown in the figure, and then twisting that solid torus into the shape of K.^[2] This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the Bing link.^[3]

See also

• Slice knot
• Seifert surface
• Satellite knot
• Whitehead double

References

Notes

1. Cimasoni 2006, p. 2395.
2. Cimasoni 2006, p. 2397.
3. Jiang et al. 2002, pp. 189–190.

Sources

• Bing, R. H. (1952), "A homeomorphism between the 3-sphere and the sum of two solid horned spheres", Annals of Mathematics, 56 (2): 354–362.
• Cimasoni, David (2006), "Slicing Bing doubles", Algebraic & Geometric Topology, 6: 2395–2415.
• Jiang, Boju; Lin, Xiao-Song; Wang, Shicheng; Wu, Ying-Qing (2002), "Achirality of knots and links", Topology and its Applications, 119 (2): 185–208, arXiv:math/9905158, doi:10.1016/S0166-8641(01)00062-1, ISSN 0166-8641.

Further reading

• Cochran, Tim; Harvey, Shelly; Leidy, Constance (2008), "Link concordance and generalized doubling operators", Algebraic & Geometric Topology, 8 (3): 1593–1646, arXiv:0801.3677, doi:10.2140/agt.2008.8.1593.
• Cha, Jae Choon; Livingston, Charles; Ruberman, Daniel (2008), "Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles", Mathematical Proceedings of the Cambridge Philosophical Society, 144 (2): 403–410, arXiv:math/0612419, doi:10.1017/S0305004107000795.