### August 2007

I’m back from Ohio, and ready to get back to the math blogging. Had my preliminary exams (written quals) today, so that should explain the delay in posting. Instead of covering anything I saw at YMC (though I will likely look at Grassmanians, Schubert Calculus, and Equivariant Schubert Calculus in the future), I’m going to talk about classical invariant theory and how to use it to solve the general degree 3 polynomials. (more…)

Ok, ok, I didn’t post anything this past week. And I won’t be getting anything in on Monday, however, I have a good reason. This weekend, I’m off to the 2007 Young Mathematician’s Conference at Ohio State. So when I get back, either I’ll do something connected to my talk (on Equivariant Cohomology and Lagrangian Grassmanians) or else if I see a really cool talk, I’ll discuss it. Stand by for a real post in the middle of next week.

So last time, I talked about the basics of polynomial knots as individual objects. Now, we’re going to talk about their parameter spaces as well as the most important new knot invariant they have given us.

So first off, we recall that a polynomial knot is given by $x(t),y(t),z(t)\in \mathbb{R}[t]$. If these are polynomials of degrees $a,b,c$ respectively, then we say that the knot defined by $(x(t),y(t),z(t))$ is of degree $\max\{a,b,c\}$. Then, the degree of a knot type is defined to be the least degree of any representation.

So I’m going to start out with something that I’ve been thinking about for about a year now, though not very hard. To begin with, a knot is an embedding of the circle into the three sphere, and for our purposes, it is important to use the 3-sphere rather than $\mathbb{R}^3$. We say that two knots are equivalent if we can manipulate one without having to break or cross strands and can obtain the other (rigorously, we say they are equivalent if their images differ by an ambient isotopy of $\mathbb{S}^3$).

Hi, I’m Charlie. I’m starting graduate school in math at the University of Pennsylvania in the fall, and I clearly have way too much time on my hands, because I’ve decided to start a math blog and, worse, to drag my friends into it as well.

I’m interested in a wide variety of mathematics, but primarily algebra, geometry and topology, specifically algebraic geometry and occasionally mathematical physics. Expect very geometric posts from me, as well as the occasional bit of category theory. My intention is to attempt to post something every Monday, and possibly more often when I come across something too interesting to sit on.