On Sun, May 24, 2020 at 9:20 AM Adam P. Goucher <apgoucher@gmx.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.
and then 'compile' that into a 4-variable real parameterisation (where the second equation is just the equation for the 3-sphere), and then stereographically project that into a 3-variable real parameterisation.
More generally, this idea will work for every torus knot.
Sent: Sunday, May 24, 2020 at 1:51 PM From: "Veit Elser" <ve10@cornell.edu> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] [xbbn] Graduate Student Solves Decades-Old Conway Knot Problem | Quanta Magazine
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com