The Conway Knot - Solved!

It was recently announced that Conway's knot problem has been solved.  This particular problem has been unsolved for about 50 years but a young graduate student successfully determined it was not a slice.

Conway's knot problem involves a knot with 11 crossings.  It was named after John Horton Conway and is similar to the Kinoshita - Terasaka knot.  The question about this knot involved  being able to slice the knot.

This problem of knot theory is actually a branch of topology.  It looks at the nature of the spaces in something resembling tangled loops.  This particular area of study has contributed to the understanding of DNA, the behavior of economic markets, to even the universe.  They've been looking at both one dimensional knots and the behavior of two dimensional knots in four dimensional space.  In mathematics a knot does not have two distinct ends, its ends are connected and cannot unravel.

The question involved with the Conway knot boiled down to "Is the Conway knot what is left after a knotted spare is sliced?"  Is the Conway knot a slice?  In other words, can it be untangled but some knots are so crumpled, they cannot.What made the Conway knot stand out is that Mathematicians were able to figure out the answer for all sorts of knots with 12 or fewer crossings except for this particular 11 crossing knot.

The grad student, Lisa Piccirillo heard about the problem back at a conference on low- dimensional  topology and geometry back in 2018.  It piqued her curiosity so she decided to work on it outside of school as if it were a homework problem.  She had a new technique in mind she wanted to try and it worked.  In about a week she had the answer and when she shared it with one of her professors who said it had to be published.

The proof is based on creating a complicated knot that is trace sibling for the Conway knot. She did this because it is known that trace siblings have the same slice status.   Then she applied the Rasmussen's s-invariate to the trace sibling and showed it was not a slice therefore the Conway knot is not smoothly slice.  This means that it is a slice of a crumpled knot but not a smooth one.  Her proof was nice and short and elegant.  It lead to her receiving and accepting an  offer from MIT for a tenure track position.

Unfortunately, John Horton Conway just passed away on April 11, 2020 from COVID-19 at the age of 82.  He contributed so much to the field of mathematics during his lifetime.  Did he learn about this solution to the Conway knot?  I don't know but I am sure if he did, he'd find it quite intriguing.  Let me know what you think, I'd love to hear.