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 . 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 ).
We can take any knot and manipulate it to get a point at infinity, and then we can just remove that point to get a curve in . We can get back to the original knot simply by adding the point at infinity back in, so we’ve lost no information. So the first thing to know is that there is a theorem by Shashtri [ Tohoku Math. J. (2) 44 (1992), no. 1, 11–17] that says that every knot is equivalent to a polynomial knot. What that means, is that we can deform our knot and get one that is parametrized by polynomials.
The first question that you might ask is “Why did Shastri care?” This is a very good question, and the answer goes back to a conjecture of Abhyankar that he made at a conference in 1977. Abhyankar asked if there exists two polynomial embeddings of into which are not equivalent under the polynomial automorphisms of . This corresponds to asking whether we can ALGEBRAICALLY untie knots. Now, certainly that’s not the case for , but what about over ? Then we can until them topologically, but the algebraic question is still open. So Shastri thought that if it is true, then a polynomial knot would make a good counterexample.
So that’s the motivation for the discussion. Now we should look at polynomial knots themselves, because studying them is just as good as studying knots, and we also have polynomials to work with. So the first question is: can these be constructed algorithmically. The answer, due to Matt (one of the other authors of this blog) in 2006, is yes! Matt’s algorithm takes an arbitrary knot parametrization as input, and will output a polynomial parametrization. The way this works is that it finds polynomials with the same crossing data, that is, that cause the knots to pass over and under in the correct sequence, given the other two parametric equations.
In fact, this algorithm gives polynomials with rational coefficients! This means that we can clear denominators and we’ve actually parametrized knots with elements of . Why is this good? Well, the best thing about polynomial knots is that they let you do all sorts of algebraic geometry things to study knots. Some examples are using Bezout’s Theorem to find relationships between degree and crossing number, finding relationships between degree and other knot invariants such as the bridge number, as well as some things I’ve been fiddling with involving singularities of plane curves and sheaves of vector spaces.
However, back to the consequences of integer parametrizations, we should keep in mind that these knots are rational curves in , something that we know a LOT about. This suggests that we can try to use reduction mod p methods to get results in knot theory, something that, to my knowledge, no one has ever considered before. We can even use this, as well as some algebraic notions of equivalence (especially if Abhyankar’s question is answered in the affirmative) to define a knot theory over an arbitrary field, and perhaps could help knot theory, algebraic geometry and number theory interact. I’m currently looking for some open problems in knot theory that I might try these methods on, if anyone has any suggestions, let me know.
I think that’s enough for this post, next time, I’ll take up the thread with spaces of polynomial knots, a paper of Vassiliev, and some more open problems.