Congratulations, Dr. Lunel!
Today I participated remotely in the successful doctoral defense of Corentin Lunel, who has been working in algorithmic low-dimensional topology with Arnaud de Mesmay and Pierre Dehornoy at the Laboratoire d’Informatique Gaspard Monge of Université Gustave Eiffel.
Corentin’s work first came to my attention at SoCG 2023, where he had a paper on the treewidth of knots and links with his advisors, later incorporated into his thesis. Treewidth is a concept in graph theory but it can be applied to knots and links by projecting them onto a plane and considering the 4-regular planar multigraphs that result. The treewidth of a knot or link is the minimum treewidth, obtained in this way, of any of its projections. It was known that this number can be arbitrarily large but Corentin provides a simpler construction: the torus knots. The illustration below is torus knot (11,10) where the first number counts the lobes and the second counts the number of strands you would have to cross to get from the center to the outside. This drawing style for these knots looks like a large grid, having high treewidth, and Corentin shows that its treewidth is within a constant factor of optimal for these knots.
The tools he uses to prove this also seem quite interesting. He defines a notion of “sphere decompositions” for three-dimensional structures, obtained by forming a family of nested double bubbles that partition space into annular shells between an enclosing bubble and the double bubble that it encloses, and topologically sweeping each of these shells by a sphere. He defines the “spherewidth” of a link to be minmax number of crossing points one of the spheres in this sweep, minimized over sphere decompositions and maximized among the spheres of the decomposition. If the treewidth of a knot or link is low, it also has a sphere decomposition of low spherewidth. But the spherewidth of the torus knots \((p,q)\) turns out to be at least proportional to \(\min(p,q)\): any lower width would allow you to attach a disk across the hole of the torus whose boundary crosses few strands of the torus knot, known to be impossible. Therefore, the treewidth is also large.
A second part of Corentin’s thesis is also related to structural graph theory in a different way. He defines a class of links obtained by “plumbing” double-twisted bands together according to a certain tree structure; this means laying one band perpendicularly across another and gluing them together where they overlap. The links are the boundaries of the resulting surfaces. He then uses Kruskal’s tree theorem to show that when a link property behave nicely with respect to cutting the surface without separating it, it can be characterized by a finite set of forbidden subsurfaces. This leads to a proof of existence of an algorithm for testing whether one of these links has bounded “genus defect”, the difference between the minimum genus of a Seifert surface and of certain four-dimensional surfaces. But although we can prove the algorithm exists, we don’t know what it is! That’s because we don’t know how to explicitly list the forbidden subsurfaces. This is all from another paper in SoCG 2024.
The thesis concludes with a third result: there exist links that are split (there exists a sphere that separates some components of the link from the rest) but for which proving this by a sequence of Reidemeister moves may have to make arbitrarily large increases in the crossing number. Both the construction (from two linked torus knots, drawn as above, and a third unknot that is split from them but not drawn as split) and the proof that the crossing number must increase use the methods from the SoCG 2023 paper. This raises the strong hope that we can prove similar results on large increases in crossing number for the unknotting problem as well.
I think these are very interesting results based on powerful new machinery that could prove very useful in continued research in this area. My understanding is that Corentin is continuing as a postdoctoral researcher in Montpellier, but he will go on the job market again soon, and should be a very strong candidate when he does.
Congratulations, Corentin!