Conway knot

Last updated
Conway knot
Conway knot.png
Braid no. 3 [1]
Hyperbolic volume 11.2191
Conway notation .−(3,2).2 [2]
Thistlethwaite 11n34
Other
hyperbolic, prime, slice (topological only), chiral
Conway knot emblem on a closed gate at Isaac Newton Institute Gateknot.jpg
Conway knot emblem on a closed gate at Isaac Newton Institute
Conway knot Conway-knot.png
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. [1]

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. 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.
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