I believe there’s a theorem in algebraic geometry that implies that those straight lines are factors of the system of polynomials, and can be removed rather than having to add a third equation. Varieties like e.g. the union of a line and a circle are not “irreducible”, and they can be factored into irreducible ones.
On May 27, 2020, at 11:30 AM, Brad Klee <bradklee@gmail.com> wrote:
Hi Dan,
Again, I think it my last message was fine as stated, but I will explain a few finer points.
Call the three polynomials S(X,Y), H(X,Y,Z), and F(X,Y,Z). They all equal zero for the parametric curve {X(t),Y(t),Z(t)} after reducing terms. The linear height function, H, can be factored as
H = L(X,Y)+C(X,Y)*Z
with L the product of four linear factors, and C the product of two circular factors. Eight solutions of L=0 & C=0, are a solutions of H=0 & S=0 regardless of Z, see:
So the algebraic varietry { (X,Y,Z) : H=0 & S=0 } accidentally contains 8 vertical lines. If (X_0,Y_0) is any one of the eight singular points, then F(X_0,Y_0,Z)=0 implies either:
48 Z^2-45 = 0, or 144 Z^2 - 567 = 0.
On the set of singularities--which corresponds to the set of knot crossings--the quadratic filter function F chooses just two points from the singular, vertical line, one on the overpass, and one on the underpass. Thus the variety:
{ (X,Y,Z) : H=0 & S=0 & F=0 }
Is the (4,2) torus knot in the strict sense, where it contains no extra points or mirror images.
--Brad
On Wed, May 27, 2020 at 11:42 AM Dan Asimov <dasimov@earthlink.net> wrote:
We need three constraints to accomplish what?
—Dan
----- Version 1.0 in Mathematica: https://0x0.st/ip7E.txt Analyzing outputs, I found the following for case (3,4):
{X,Y,Z} = {Sin[t] + 2 Sin[3 t], Cos[t] - 2 Cos[3 t], 2 Sin[4 t]} 0 = -81 + 117 X^2 - 40 X^4 + 4 X^6 + 117 Y^2 - 112 X^2 Y^2 + 12 X^4 Y^2 - 40 Y^4 + 12 X^2 Y^4 + 4 Y^6; 0 = -8 X^3 Y + 8 X Y^3 + 27 Z - 24 X^2 Z + 4 X^4 Z - 24 Y^2 Z + 8 X^2 Y^2 Z + 4 Y^4 Z;
0 = 81 - 81 X^2 - 81 Y^2 + 32 X^2 Y^2 + 16 X^2 Z^2 + 16 Y^2 Z^2
Contrary to expectation, we need not two but three constraints. The first two equations define the knot + 8 vertical lines at crossing points. The third constraint filters out vertical lines. -----
_______________________________________________ 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
Cristopher Moore Professor, Santa Fe Institute I can think of nothing more dangerous, more divisive, or more self-destructive than the effort to prey on what is called 'white backlash'... it is dangerous because it threatens to vest power in the hands of second-rate men whose only qualification is their ability to pander to other men's fears. I think it divides this nation at a very critical time—and therefore it weakens us as a united country. — Lyndon Johnson, 1966