> I'm sure most of you know more about this than me. But since I'm in a kind > of pseudo-holiday state between work and doing nothing, perhaps you are too: > > Amazing: Karim Adiprasito proved the g-conjecture for spheres! > https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/
Huh. I saw the abstract as I was perusing math/new at the ArXiV yesterday night (mostly looking for math.GT, Geometric Topology being my general field, and knot theory my specialty) and declined to download the paper. Now that I've read the blog post there I may have to reconsider; while I appreciate that computability classes are important, generally speaking I am not personally interested in computing knot invariants (efficiently or otherwise); I'm more interested in discovering them, and relating them to each other. My article in the Handbook of Knot Theory (Elsevier, 2004) has a bit of (entirely justifiable!) snark near the beginning, attached as a footnote to a few sentences that I think still provide a useful distinction (or three): ==== In the past several decades, knot theory in general has seen much progress and many changes. "Classical knot theory"-the study of knots as objects in their own right-has taken great strides, documented throughout this Handbook [blah, blah]. Simultaneously, there have been extraordinarily wide and deep developments in what might be called "modern knot theory": the study of knots and links in the presence of extra structure, for instance, [blah blah] [footnote] Some observers have also detected "postmodern knot theory": the study of extra structure in the absence of knots. ==== What I do is "modern knot theory". But so is this new paper (and the various older papers one discovers by clicking links in the blog post), and as such I'm all for it. On the other hand, _even if_ detecting the unknot (say) is in P (and even if, also, P equals NP), that doesn't mean you'll ever be able to *do* it (decide whether a particular knot diagram is a diagram of the unknot) for every knot that you might want to, no matter how fast computers get and how efficient those polynomial-time/polynomial-space unknot detection algorithms get: because "every knot that you might want to", if your wants are like mine, will always include _infinite families of knots_. I like theorems of the form "if X belongs to (infinite family) F, then X has property G." (By that token, I should, and do, like theorems that say such-and- such is not algorithmically decideable at all! For instance, it's quite comforting, in its way, to know that there's no algorithm that can tell you, given any two four-dimensional closed manifolds, whether or not they are the same; and likewise, that there *are* such algorithms for three-dimensional closed manifolds.) But there's still plenty of post-modern knot theory out there, and I still don't approve of it. ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives back to 2003: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/ by Dr. Strangelove
