That's a much neater equation than I expected! I wonder if there's a nice parametric form using elliptic functions à la https://arxiv.org/abs/1501.07157 (see e.g. Lemma 3.11)
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Replying to @blockspins @JDHamkins
Well, one can compute the curve's genus. I'm a bit too tired to attempt this right now. Another thing would be to understand why the equation given by elimination theory had another component (a circle with center (0,−1) and radius 2): this is probably obvious, but IDC.
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Replying to @gro_tsen @JDHamkins
The other component arises if you "flip" the smallest rhombus, collapsing the blue point to one of the other joints.pic.twitter.com/Jaew3aF7Vo
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And unless I made a typo, the curve seems to be genus 3, unfortunately: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens() qc = QuarticCurve(x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4) qc.genus() ## result: 3
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But now I'm a bit confused (and out of my depth): qc.geometric_genus() ## result: 0
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Replying to @blockspins @JDHamkins
Yes, that's the sort of thing I feared: there's probably a lot of fine print in Sage's genus computing commands about what it computes exactly and how. (Me, I can't even ever remember which is which between geometric and arithmetic genus, so…)
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Replying to @gro_tsen @JDHamkins
I believe the geometric genus is the one that matters re: parametrization; for some reason the QuarticCurve.genus() method gives the arithmetic genus (contradicting the behavior of Curve.genus(), of course...). Anyways, it turns out Sage has Curve.rational_parametrization() ...
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And the output is: Scheme morphism: From: Projective Space of dimension 1 over Rational Field To: Projective Plane Curve over Rational Field defined by x^4 + 2*x^2*y^2 + y^4 + 10*x^2*z^2 - 6*y^2*z^2 + 8*y*z^3 - 3*z^4 ...
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Defn: Defined on coordinates by sending (s : t) to (54*s^4 - 36*s^2*t^2 - 16*s*t^3 - 2*t^4 : -63*s^4 - 84*s^3*t - 26*s^2*t^2 - 20*s*t^3 + t^4 : 45*s^4 - 12*s^3*t + 22*s^2*t^2 + 4*s*t^3 + 5*t^4)
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A final picture: parametric_plot( (2*(27 - 18*t^2 - 8*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4), -(63 + 84*t + 26*t^2 + 20*t^3 - t^4)/(45 - 12*t + 22*t^2 + 4*t^3 + 5*t^4) ), (t,-1e5,1e5), plot_points=1e6)pic.twitter.com/2Wt2w12BBE
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Just for fun, here's a "conformal neighborhood" of the curve. The small gap in the upper left is where |t|>100. I'm impressed that you guys could come up with both implicit and parametric formulae for the curve. I didn't know the latter was even possible.pic.twitter.com/r0pH4tRMSL
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We got lucky, I think. The algebraic curve just happened to be genus 0 which means that it's secretly a (projective) line in disguise: https://en.wikipedia.org/wiki/Algebraic_curve#Rational_curves … I do wonder whether we could have figured out the genus from the linkage or some other way without relying on Sage.
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Replying to @blockspins @peterliepa and
Maybe there's an argument that uses the combinatorics of the linkage to get the degree of the polynomial in x&y; then we observe from geometry that one irreducible component is a circle (the "collapsed" picture above) and that what's left over must be a quartic plane curve...
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