Conway knot

Conway knot
Braid no.3
Hyperbolic volume11.2191
Conway notation.−(3,2).2
Thistlethwaite11n34
Other
hyperbolic, prime, slice (topological only), chiral
Conway knot emblem on a closed gate at Isaac Newton Institute
Conway knot
Conway knot
A five star version of the Conway knot

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 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 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. 1 2 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. 1 2 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.