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 . If these are polynomials of degrees respectively, then we say that the knot defined by is of degree . Then, the degree of a knot type is defined to be the least degree of any representation.
So for example, Shastri’s original paper included a parametrization of the trefoil knot, which is of degree 5. It happens that this is the lowest possible degree for the trefoil. He also included a degree 6 figure eight knot, which is the lowest possible degree as well.
Now, we will define the space of knots of degree . Let be polynomials such that at least one has degree and the others are of at most degree . The space of such polynomials is , because a degree $d$ polynomial has coefficients. Well, technically this isn’t QUITE right, because this includes all polynomials of lower degree as well, but we shall correct this by removing the subset of triples of polynomials that (i) have no terms of degree , (ii) have a cusp, that is, there exists a such that , or (iii) have a crossing, that is, there exist such that . This gives us , the space of three dimensional polynomial knots of degree . The stands for Vassiliev, and we will get to him in a moment.
Note that over the complex numbers, resultants can be used to find polynomial equations for the locus of such points. However, because we only want to ignore real cusps and real crossings, we have a problem and these points are a bit harder to work out. Now, we can state a few theorems that are not completely trivial. (as a general reference, see Don O’Shea and Alan Durfee’s paper on the Arxiv)
So the first result is that if two polynomial knots in the same connected component of , then they are equivalent as knots. The proof requires differential geometry, though somehow it seems that there should be an elementary or at least algebraic proof, and finding one would be nice. But immediately one must ask, “What about the converse?” Well, this is open. It is not known if there is always at most one component of equivalent knots.
To investigate this question, we need to be able to determine things like what the minimal degree in which a knot can appear is. It has been shown that once a knot appears, it appears in all greater degrees, so the question of finding all degrees in which a knot appears is just that of finding the lowest. There are a few tricks to this. The first is what is called a monoidal transformation. These are transformations of of the form or the obvious analogs for the other variables. This can decrease the degree of the parametrization sometimes, but only when the degree of can be written as a positive combination of the degrees of and . This is extremely valuable, because it doesn’t change the parametrization much, it merely removes superfluous terms.
A somewhat more interesting method is to combine monoidal transformations with the knot perturbation trick. This trick, which is also the one that proves that knots exist in all degrees higher than the ones they’ve been discovered in, allows the addition of a higher degree term to any of the polynomials parametrizing the knot…albeit one multiplied by a sufficiently small number. The point of this is to change the degrees of two of the polynomials so that they can be used to remove terms from the greater one.
For instance, the knot was known to be of degree at most 11 before the perturbation trick was worked out, and there was no possible monoidal transformation to lower the degree. However, using the trick and a monoidal transformation, the degree was reduced to 9. This merely gives lower degree examples, though. It cannot prove that a knot exists in no lower degree.
Also, a use of Bezout’s theorem can give some crude bounds on the crossing number. If and , then it says that the crossing number is at most . So, in terms of degree, this is . So then the knot may exist in degree as low as 6, though no parametrization of such low degree is known. In fact, it is possible that all knows of crossing number up to 6 exist in degree 6, but this is not known.
This question can be answered by analyzing more carefully. Vassiliev has analyzed it here though I must admit that I don’t understand his methods very well, so I have been unable to attempt to extend them to checking out . But merely knowing the number of connected components would be a step towards solving two open problems: are there any knots of crossing number 5 or 6 in degree 6, and are there any knots of degree 6 with multiple path components?
That’s all I’m going to say about polynomial knots for awhile, there are so many open problems in the area that it’s a shame that so few people know about it. Anyway, next week I’ll talk about something else entirely…though I haven’t decided what yet.