Here’s a graduate-student-level question about knots — the deepest I’m capable of: Is it possible to define a “real-algebraic degree” for knots, and if so, has it been tabulated? My idea is to first construct knots as non-singular algebraic varieties, i.e. as the solution set of a pair of polynomial equations in x,y,z, and I’m using non-singular to mean that locally, near every point of the solution set, the linear approximation of the variety is two planes intersecting as a line. For example, x^2+y^2=1 z=0 defines the unknot. The variety defined by the pair of equations will in general have several disconnected components. We only care if *one* of these components is our target knot, and if so, we take some suitable “minimal” combination of the degrees of the two polynomials as the “real-algebraic degree” of our knot, e.g. min(d1+d2), min(max(d1,d2)). By the second definition, the unknot would have degree 2. Two specific questions: 1) Can all knots be constructed in this way? 2) What’s the degree of the trefoil? -Veit
On May 24, 2020, at 1:07 AM, Brad Klee <bradklee@gmail.com> wrote:
The best way to celebrate a new result is to learn what it means for yourself. If there is animus here, it’s toward ignorance, including my own. Thanks Allan for the reference pointer, I will look into it and see if it says anything about knot isomorphism classes. I have a lot of catching up to do. Even if I can’t learn the new proof, maybe I can learn something. —Brad