On May 25, 2020, at 9:38 AM, Andy Latto <andy.latto@pobox.com> wrote:
For the trefoil, one way to construct the knot is to take the 2-variable *complex* parameterisation:
a^2 = b^3, |a|^2 + |b|^2 = 1
I think you want |a|^2 = |b|^2 = 1/2 (a torus) rather than |a|^2 + |b|^2 = 1 (a 3-sphere). It makes sense that to define a 1-dimensional curve in C^2, which is R^4 with some extra structure, we need 3 real equations, not 2.
But to get this to be an algebraic curve embedded in R^3, we need to specify a polynomial map from the torus |a|^2 = |b|^2 = 1 to a torus in 3-space. That's where you use sterographic projection to map the 3-sphere |a|^2 + |b|^2 = 1 to R^3, using the fact that the torus |a|^2 = |b|^2 = 1 lies in the 3-sphere |a|^2 + |b|^2 = 1.
Andy, I think Adam’s embedding works and I like it. Let a=r exp(i p), b=s exp(i q), where r and s are positive. Then a^2=b^3 requires r=s (=1/sqrt(2) by the 3-sphere constraint), and 2p=3q mod 2pi. That’s the curve that winds around the torus trefoil-like. But Adam’s statement "More generally, this idea will work for every torus knot.” is intriguing. It suggests that f(a,b)=0, |a|^2+|b|^2=1 where f is a polynomial, might fall short of constructing all knots. Does the set of knots constructible by this route have a name? Is this subset easier to classify than general knots? Long ago I thought the best way to classify knots was to do so within each genus. After all, genus-1 knots (torus knots) are very straightforward. But it seems the complexity skyrockets already at genus-2! I like the algebraic approach, like Adam’s, because it opens the possibility of a natural taxonomy. Consider the space defined by the monomial coefficients, which is finite dimensional if we bound the degree. This space has singular loci where one knot/link combination transforms into another. Reidemeister moves also give you a taxonomy, but the algebraic scheme may be more “natural”. -Veit