There’s a lovely picture on the cover of the Proceedings of the National Academy of Sciences this week:
And a description of what it depicts, which is intriguing:
Cover image: Pictured is a modern version of the Borromean rings, a topological arrangement of three interlocked symmetric rings that owes its name to the Borromeo family of Italy on whose coat of arms the rings appear. Although the three rings cannot be pulled apart, no two of them are linked—a fact that becomes apparent when one of the rings is hidden from view. Jim Conant, Rob Schneiderman, and Peter Teichner derived this particular realization of the link from their theory of Whitney towers, where it represents the Jacobi identity, or IHX-relation. See the article by Conant et al. on pages 8131–8138, which is part of the Special Feature on Low Dimensional Geometry and Topology. Image courtesy of Jim Conant, Rob Schneiderman, and Peter Teichner.
But when you go to the paper, you’ll see that its abstract is so opaque to a non-mathematician that it might as well be written in Martian:
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
(Presumably “Arf invariants” don’t refer to the unchanging vocalizations of a dog. )
This shows how far removed mathematics is from even other scientists. Or are our own biology abstracts just as opaque to mathematicians?
Conant, J., R. Schneiderman, and P. Teichner. 2011. Higher order dimensions in low level topology. Proc. Nat. Acad. Sci. USA 108:8131-8138.
h/t: Matthew Cobb