Bing double

{{Short description|Operation on a knot producing a link with two components}}

[[File:Bing double of unknot.svg|thumb|The unknot (left) and its Bing double (right).]]

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 {{harvtxt|Bing|1952}} by their namesake, the American mathematician R. H. Bing.{{sfn|Cimasoni|2006|p=2395}} The Bing double of a slice knot is a slice link, though it is unknown whether the converse is true.{{sfn|Cimasoni|2006|p=2397}} The components of a Bing double bound disjoint Seifert surfaces.{{sfn|Cimasoni|2006|p=2397}}

[[File:Solid torus surrounding Bing double of unknot.svg|thumb|A solid torus encasing the Bing double of the unknot. ]]

The Bing double of a knot {{math|''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 {{math|''K''}}.{{sfn|Cimasoni|2006|p=2397}} This definition is similar to that for Whitehead doubles. The Bing double of the unknot is also called the '''Bing link'''.{{sfn|Jiang|Lin|Wang|Wu|2002|pp=189-190}}

== See also == * Slice knot * Seifert surface * Satellite knot * Whitehead double

== References ==

=== Notes === {{reflist}}

=== Sources === *{{citation |last=Bing |first=R. H. |title=A homeomorphism between the 3-sphere and the sum of two solid horned spheres |journal=Annals of Mathematics |year=1952 |pages=354–362 |volume=56 |number=2 }}. *{{citation |last=Cimasoni |first=David |title=Slicing Bing doubles |journal=Algebraic & Geometric Topology |year=2006 |pages=2395–2415 |volume=6 }}. *{{citation |last1=Jiang |first1=Boju |last2=Lin |first2=Xiao-Song |last3=Wang |first3=Shicheng |last4=Wu |first4=Ying-Qing |title=Achirality of knots and links |journal=Topology and its Applications |volume=119 |issue=2 |year=2002 |pages=185-208 |issn=0166-8641 |doi=10.1016/S0166-8641(01)00062-1 |arxiv=math/9905158 }}.

==Further reading== *{{citation |last1=Cochran |first1=Tim |last2=Harvey |first2=Shelly |last3=Leidy |first3=Constance |title=Link concordance and generalized doubling operators |journal=Algebraic & Geometric Topology |volume=8 |issue=3 |year=2008 |pages=1593–1646 |doi=10.2140/agt.2008.8.1593 |arxiv=0801.3677 }}. *{{citation |last1=Cha |first1=Jae Choon |last2=Livingston |first2=Charles |last3=Ruberman |first3=Daniel |title=Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=144 |issue=2 |year=2008 |pages=403-410 |doi=10.1017/S0305004107000795 |arxiv=math/0612419 }}.

Category:Knot theory