Conway knot

The Conway knot is an 11 crossover knot named after mathematician John Horton Conway . The knot has the braid word . As is usual with mathematical knots, the ends of the cords in the Conway knot are not loose, but rather glued together, and they form a self-contained construct. ${\ displaystyle \ sigma _ {2} ^ {3} \ sigma _ {1} \ sigma _ {3} ^ {- 1} \ sigma _ {2} ^ {- 2} \ sigma _ {1} \ sigma _ {2} ^ {- 1} \ sigma _ {1} \ sigma _ {3} ^ {- 1}}$

For a long time it was unknown whether the knot was a so-called smooth disc knot . In March 2020, an article by mathematician Lisa Piccirillo was published in the Annals of Mathematics proving that he is not.

Mathematically, it is known that if two nodes have the same four-dimensional form called a trace (a 4-manifold assigned to the node), they are either both disc nodes or both are not disc nodes. Piccirillo solved the math puzzle by constructing a second knot, the Piccirillo knot, which had the same trace as the Conway knot. Since she was able to prove that her knot is not a disk knot, she was also able to prove that the Conway knot is not a disk knot.

literature

Conway's Knot (Wolfram MathWorld, accessed June 4, 2020)
Knot Theory - Graduate Student Solves Decades-Old Conway Knot Problem (Quanta Magazine, accessed June 4, 2020)
Graduate Student Untangles Decades-Old Math Problem in Less Than a Week (Smart News, accessed June 4, 2020)
PhD student solves important math puzzle - US mathematician Lisa Piccirillo finds the solution to the Conway node (Süddeutsche Zeitung, June 2, 2020, 6:52 p.m.)
Lisa Piccirillo: The Conway knot is not slice. Annals of Mathematics. Vol. 191, No. 2 (March 2020), pp. 581-591 (JSTOR); Preprint on arxiv.org (August 8, 2018)