A few days ago, two people named Joshua (one Howie and one Greene) independently posted to arXiv a similar solution to an old question of Ralph Fox:
Question: What is an alternating knot?
The preprints are:
This post will briefly introduce the problem; I look forward to reading the solutions themselves!
An alternating knot diagram is a diagram of a knot in which crossings alternate over and under along the knot. A knot is alternating if it has an alternating diagram. The Tait conjectures are about properties of alternating knots, and there are various other theorems in topology, e.g. in Floer homology, in which alternating knots are special. Given that a knot diagram is a mere combinatorial shadow of a topological object (a knot in 3-space), it seems that there ought to be a simple topological characterization of alternating knots in 3-space, with no mention of diagrams. Unfortunately there has not been such a characterization… until now!
Josh G.’s characterization works in a more general setting than Josh H.’s, but they both come down to the same idea- properties of pairings on the first homologies of checkerboard surfaces. In Josh G.’s case (which works also for links), there is a pairing on the first homology of a surface in a homology sphere, and the checkerboard surfaces of the knot are isotopic rel. boundary to two surfaces, one of which has a positive definite pairing, and the other a negative definite pairing.
This is so simple and elegant!! How could it take so long to discover ??
Both solutions yield very similar exponential-time algorithms to determine whether or not a knot is alternating (this would be much harder via the diagram).
The results of the two Joshuas might open the door to a geometric topological proof of Tait’s flyping conjecture. This would very nicely steal the thunder of quantum topology- the great classical triumph of quantum topology has always been the proof of the Tait conjectures, which had no other known proof.
Thanks to Dave Futer for calling my attention to these preprints!