In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway.[1]
| Conway knot | |
|---|---|
 | |
| Braid no. | 3[1] |
| Hyperbolic volume | 11.2191 |
| Conway notation | .−(3,2).2[2] |
| Thistlethwaite | 11n34 |
| Other | |
| hyperbolic, prime, slice (topological only), chiral | |
It is related by mutation to the Kinoshita–Terasaka knot,[3] with which it shares the same Jones polynomial.[4][5] Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot.[6]
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.[6][7][8] 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).[9]
References
edit- ^ a b Weisstein, Eric W. "Conway's Knot". mathworld.wolfram.com. Retrieved 2020-05-19.
- ^ 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.
- ^ Chmutov, Sergei (2007). "Mutant Knots" (PDF). Archived (PDF) from the original on 2016-12-16.
- ^ Kauffman, Louis H. "KNOTS". homepages.math.uic.edu. Retrieved 2020-06-09.
- ^ 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.
- ^ a b 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.
- ^ 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.
- ^ Klarreich, Erica. "Graduate Student Solves Decades-Old Conway Knot Problem". Quanta Magazine. Retrieved 2020-05-19.
- ^ Klarreich, Erica. "In a Single Measure, Invariants Capture the Essence of Math Objects". Quanta Magazine. Retrieved 2020-06-08.
External links
edit- 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.