Conway knot
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.[2]
Conway knot | |
---|---|
Crossing no. | 11 |
Genus | 3 |
Conway notation | .-(3,2).2[1] |
Thistlethwaite | 11n34 |
Other | |
, prime |
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
- 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.
- Weisstein, Eric W. "Conway's Knot". mathworld.wolfram.com. Retrieved 2020-05-19.
- Chmutov, Sergei (2007). "Mutant Knots" (PDF).
- 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. Retrieved 2020-06-09.
- 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
- Conway knot on The Knot Atlas.
- Conway knot illustrated by knotilus.
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