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
Genus3
Conway notation.−(3,2).2[1]
Thistlethwaite11n34
Other
prime, prime
Conway knot emblem on a closed gate at Isaac Newton Institute.
Conway knot
Conway knot

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
  1. 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.
  2. Weisstein, Eric W. "Conway's Knot". mathworld.wolfram.com. Retrieved 2020-05-19.
  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.
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