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!