At the risk of embarrassing myself, here’s a linear algebra question that has turned up in a knot theory project I’m working on. The post might be embarrassing because the answer might be “obviously not,” and I just don’t have enough linear algebra intuition to see it. On the other hand, this would contradict some things which seem to make sense from a knot theory perspective.

Say we have an matrix . We can assume throughout that is invertible, though maybe it doesn’t matter. Its determinant has the expansion

where is the symmetric group on objects. One nice way to think of this (which might help for the problem), is that a permutation corresponds to a “choice” of a column of by each row of (so row chooses column ), amounting to a particular bijection between rows and columns. Then, in the summand corresponding to in the above expansion, we see the element of each row which lies in the column it chose. (I’ve just learned that this way of looking at the expansion is exactly how Kauffman came up with his states-sum model for the Alexander polynomial. Would anyone be interested in an intro-post on the Alexander polynomial?)

Now suppose we have a matrix whose entries are only 0’s and 1’s. Suppose, furthermore, that all non-zero summands in the above determinant expansion are , i.e., the corresponding permutation is even. Call such a matrix special. Then the set of bijections between rows and columns which contribute a non-zero amount to the expansion has cardinality the determinant; call this set S1. First question: how to characterize special matrices in some less convoluted way (this is a side-question, but if there were some nice geometric characterization, it might help with the other question).

Now, another set which has cardinality is the set S2 of solutions to , where is some vector in . To see this, just consider —it contains exactly integral lattice points, and their preimages under are the solutions in question.

So, my big question: is there some natural bijection between S1 and S2, some way to go from a bijection between rows and columns, to a particular solution to the system of equations over ? As I said at the beginning, I’m not sure whether there should be. It seems to me that maybe one just has to sit down and explicitly translate from the algebraic to the geometric definition of determinant. But I told myself I would abandon this project until the end of the summer, so maybe someone else wants to have a go? I’d appreciate any insights! Another option is that maybe there is some kind of “affine” bijection: we can get from one permutation to another by transpositions, and maybe there is some corresponding way to go between solutions?

In a later post, maybe I’ll talk about the knot theory part of this. On one side we have binary-dihedral representations of a knot group, and on the other side we have Kauffman states, or spanning trees of a certain graph, if you like, which happen to generate both the Khovanov and knot Floer complexes. The “Alexander matrices” which show up are special, in the above sense, in the case of alternating knots.

Thanks for any help!

On Friday, January 30th, Concrete Nonsense will host the 48th Carnival of Mathematics. So if you’ve ever wanted to be one of the carnival folk, now’s your chance: submissions are welcome! Any type of math, at any level, in any context. See the two most recent Carnivals here and here.

Here’s how it works: You post your submission in your own web space, and the Carnival links to it. That’s it. If you’re interested (or if you’re not interested but are susceptible to coercion), email us. Our email addresses are on the About Page, or you can reach me at (my last name at math.columbia.edu).

As the changes to my About post show, I am now a graduate student at MIT. The transition time (all of last month) was very hectic, so I have been lazy about posting. But a post will follow this shortly.

-Yan

Hello internet! My name is Sam (Lewallen,) and I’m the third member of this intrepid triumvirate. I am interested in everything, though my specialized knowledge is limited basically to topology and geometry, knot theory in particular. I’ve worked on Khovanov homology, and the Volume Conjecture. I’m also hugely interested in neuroscience, i.e., figuring out how the brain works (ooooh consciousness!), and I would love if fancy math ends up playing a role.

I have lots of posts planned. Here are a few:

1. Alex and I want to do a series on zeta functions, the different types that there are and how they show up everywhere.
2. I, like everyone, am fascinated by the moments when natural numbers pop up in different places, out of the blue. For example, the fact that there are 5 convex regular polyhedra (why 5?!). I want to write a series of posts showing that ALL such “arbitrary” natural numbers can be derived from this 5! Less ambitiously, I want do describe as many situations as possible where this can be done. Any ideas? Here are two: The McKay correspondence says # of polyhedra => number of families of simple Lie algebras; I also think # of polyhedra => # of divsion algebras over . Which then says something about the number of generalized cohomologies you can have, according to some talk by Peter Teichner…
3. I have lots of obscure knot theory ideas I want to write about. I want to do overviews of the Alexander and Jones polynomials (and all their various definitions), and discuss some neat ways that they’re related, one, for example, involving random walks on diagrams and zeta functions (#1! #1!)

Do any of these interest anybody? Any other suggestions?

As for my personal info, I’m going to be a G1 at Princeton next year. Anyone gonna be in the area?

Goodbye, dear internet, for now.

I don’t enjoy studying for exams very much. Rather than despair, however, I channeled my energy into a creative outlet:

300 meets the Bianchi identities: the first “No Weil!” Comics production, here.

Can you spot all four versions of the Bianchi identity I used?!?

Hey.

I’m Yan, currently 21 and in a life-long love-hate relationship with mathematics.

My main mathematical interests are problem-solving and connections between seemingly disparate fields. I have a sweet tooth for problems with short, elegant solutions. My main mathematical peeve is mathematical snobbery.

I strongly believe that there are two main types of value in mathematics – robust theories have depth and utility; elegant results have artistry and inspiration. While I’m wired to enjoy the latter more, I have great respect for those who partake mainly in the former, mostly because I lack the forms of intelligence and patience required to be a good theory builder.

Of course, my main weakness is talking too much when I could be doing something. So let’s do it.

-YZ

In real life, Yan is a dilettante too high on “useless activities” like movies or games to become a good mathematician, but he tries. His biggest hobby is talking to strangers, so e-mail him. His personal blog is Blue Slate. Yan is currently a arbitrage strategist / researcher at a hedge fund graduate student at MIT.

Why am I writing for a math blog, when I can’t even motivate myself to work on the math essay I’m supposed to be working on? This is just one of the many mysteries I’ll delve into on Concrete Nonsense.

I’m currently spending a year at Cambridge doing Part III; next fall, I’ll be starting a PhD at Columbia. My mathematical interests jump around a lot, but “geometry, topology, algebra, and physics” would be a fair start at a description. In my first “real” post, however, I’ll be talking about some recreational math(s) instead—stay tuned!

You can call me APE (my initials) for short.

We are Alex Ellis, Sam Lewallen, and Yan Zhang, three guys who studied math together as undergrads. About a year later, now living in three different cities spread over two continents, we’ve decided to start a group math blog—welcome to Concrete Nonsense. We aren’t too sure what we’ll be writing here, but expect some real variety in style and in content between the three of us.

Each of the three of us is writing an introductory post, giving some idea what we’re interested in and what we might be writing about. Also see the blog’s about page.

And in case you were wondering, the source image for our title banner is stolen (with kind permission from Grayson Sigler, the sculptor) from this post on Tracy Sigler’s blog. We fell in love with the image at first sight. Check out his blog (especially if you are a music fan)!