Misplaced Pages

Conway knot
Braid no.	3
Hyperbolic volume	11.2191
Conway notation	.−(3,2).2
Thistlethwaite	11n34
Other
hyperbolic, prime, slice (topological only), chiral

In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway.

It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot.

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

1. ^ Weisstein, Eric W. "Conway's Knot". mathworld.wolfram.com. Retrieved 2020-05-19.
2. Riley, Robert (1971). "Homomorphisms of Knot Groups on Finite Groups". Mathematics of Computation. 25 (115): 603–619. doi:10.1090/S0025-5718-1971-0295332-4.
3. Chmutov, Sergei (2007). "Mutant Knots" (PDF). Archived (PDF) from the original on 2016-12-16.
4. Kauffman, Louis H. "KNOTS". homepages.math.uic.edu. Retrieved 2020-06-09.
5. Litjens, Bart (August 16, 2011). "Knot theory and the Alexander polynomial" (PDF). esc.fnwi.uva.nl. p. 12. Archived (PDF) from the original on 2020-06-09. Retrieved 2020-06-09.
6. ^ Piccirillo, Lisa (2020). "The Conway knot is not slice". Annals of Mathematics. 191 (2): 581–591. doi:10.4007/annals.2020.191.2.5. JSTOR 10.4007/annals.2020.191.2.5.
7. Wolfson, John. "A math problem stumped experts for 50 years. This grad student from Maine solved it in days". Boston Globe Magazine. Retrieved 2020-08-24.
8. Klarreich, Erica. "Graduate Student Solves Decades-Old Conway Knot Problem". Quanta Magazine. Retrieved 2020-05-19.
9. Klarreich, Erica. "In a Single Measure, Invariants Capture the Essence of Math Objects". Quanta Magazine. Retrieved 2020-06-08.

External links

• Conway knot on The Knot Atlas.
• Conway knot Archived 2020-06-27 at the Wayback Machine illustrated by knotilus Archived 2020-06-27 at the Wayback Machine.

• Knot theory
• Non-alternating knots and links
• Hyperbolic knots and links
• Unfibered knots and links
• Prime knots and links
• Slice knots and links
• Chiral knots and links
• Non-tricolorable knots and links
• John Horton Conway