This is a nice problem and solution. There is a very fun connection with Riemann surfaces of elliptic curves once we identify C^2~R^4. So I spent a couple of hours drawing a (3,4) torus knot in Harold Edwards's normal form: https://0x0.st/ipSo.png https://0x0.st/ipSi.png https://0x0.st/ipS-.png When you have an algebraic definition, then you can integrate the knot to obtain a complex period. For the integral you don't necessarily need an algebraic definition of the knot itself. The dimensions of the period rectangle can be calculated on algebraic cycles. The knot takes a periodic trajectory through the complex, doubly-period plane of time, so its period can be counted as a pair of integers and then multiplied by the scale factor. The knot depicted above has period (3*4+4*2*i)*K(1/2), where K is the complete elliptic integral of the first kind. So, there it is, a congratulations to Lisa Piccirillo! I couldn't understand the proof, but it did, in some way, inspire me to integrate a knot period. --Brad PS. For more info on the calculation above, refer to Section IV of "An Alternative Theory of Simple Pendulum libration": https://github.com/bradklee/Dissertation/blob/master/SimplePendulum/SimplePe... On Sun, May 24, 2020 at 8:20 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
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
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