mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in particular in

knot theory In the mathematical field of 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 ...

, the Conway knot (or Conway's knot) is a particular

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

with 11 crossings, named after

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

. It is related by

mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...

to the

Kinoshita–Terasaka knot In knot theory, the Kinoshita–Terasaka knot is a particular prime knot. It has 11 crossings. The Kinoshita–Terasaka knot has a variety of interesting mathematical properties. It is related by mutation to the Conway knot, with which it share ...

, with which it shares the same

Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...

. Both knots also have the curious property of having the same

Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...

and Conway polynomial as the

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

. The issue of the sliceness of the Conway knot was resolved in 2020 by Lisa Piccirillo, 50 years after John Horton Conway first proposed the knot. Her proof made use of Rasmussen's s-invariant, and showed that the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both).

References

External links

Conway knot
on

The Knot Atlas ''The Knot Atlas'' is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan, who maintains the current site with Scott Morrison. According to Schiller, the si ...

.
Conway knot
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knotilus
Prime knots and links John Horton Conway