Borromean rings

Borromean rings
L6a4
Crossing no.	6
Hyperbolic volume	7.327724753
Stick no.	9
Conway notation	.1
A–B notation	63 2
Thistlethwaite	L6a4
Other
alternating, hyperbolic

In mathematics, the Borromean rings^[a] are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as an element of their coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan. They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative name Ballantine rings. Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the Efimov state and Borromean nuclei, both of which have three components bound to each other although no two of them are bound.

Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In knot theory, the Borromean rings can be proved to be linked by counting their Fox n-colorings. As links, they are Brunnian, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous linking properties to the Borromean rings.

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