The Unapologetic Mathematician

Mathematics for the interested outsider

Slides for Bracket Extension talk

I’ve posted the slides for my talk. There are a few typos that I noticed as I was speaking, but nothing that makes it incomprehensible. I actually started the lecture on page 10 to save time, and because the run-up is pretty standard material.

March 17, 2007 Posted by | Knot theory | 1 Comment

AMS Sectional, Day 2

Still no wireless, so I’ll again jot a little something about the noteworthy talks.

Louis Kauffman gave a talk about an invariant of “virtual” knots and links, which are described in his paper that I linked to the other day. The invariant is an extension of the Kauffman bracket that I extend to tangles. An obvious question is how to do both extensions, getting functors on the category of virtual tangles.

Heather Dye, Kauffman’s former student, then spoke on virtual homotopy. This may well be related to Allison Henrich’s work on Legendrian virtual knots. It’s all tangled (har) up together.

I gave my talk after that. I’ll make a separate post with the link to my slides.

After lunch, Alexander Shumakovitch gave a very clear (though not yet complete) combinatorial categorification of HOMFLY evaluations. There’s a parameter n in his theory, and setting it to 2 gives back the combinatorial version of Khovanov homology. Setting it to higher values should correspond to what Josh Sussan — currently finishing his Ph.D. here at Yale — has done in the representation-theoretic picture for U_q({\mathfrak sl}_n).

The last talk that really grabbed me was Michael “Cap” Khoury’s explanation of a new definition for the Alexander-Conway polynomial. The really interesting thing here is that it really looks like he’s realizing it as some sort of representable functor on some sort of category. Almost, but not quite. We talked for a bit about what’s missing, and I don’t think it’s impossible to push it a bit and get that last lousy point.

March 17, 2007 Posted by | Uncategorized | Leave a comment

Joyce on Geometry

Beannachtai na Feile Padraig.

In honor of the day, I’d like to post a passage of Finnegans Wake. There’s a beautiful section from pages 293 to 299 “explaining” Euclid I.1. It’s as good a place as any to start in on “the book of Doublends Jined”, so if this sort of thing intrigues you I hope you’ll get a copy and go from here. If nothing else I hope that my own exegeses are somewhat easier to follow.

This area of the text is particularly… texty. I’ll do my best to match the original as exactly as possible. Individual pages will be separated by hard rules. The passage itself follows the jump.
Continue reading

March 17, 2007 Posted by | Uncategorized | Leave a comment

New result

Never believe anyone who says that drinking never helps anything, for tonight over many a pint o’ Guinness I hit upon the solution to a problem that’s been nagging at me for some time. The answer is so incredibly simple that I’m feeling stupid for not thinking of it before. So here it is:

Cospans in the comma category of quandles over a given quandle Q from the free quandle on m letters to the free quandle on n letters categorify the extension of link colorings by Q to tangles.

This gives me a new marker to aim at. I can explain this, but it will require more preliminaries before it’s really accessible to the GILA (Generally Interested Lay Audience) as yet. Those who know quandles — a topic I’ll be covering early next week — and category theory and some knot theory should be able to piece together the meaning now. For the rest of you.. stay tuned.

March 17, 2007 Posted by | Knot theory | 1 Comment