Algebraic Family of Trefoil Curves
The trefoil is the simplest nontrivial knot and the only knot with three crossings . In this Demonstration the trefoil is drawn with cyclic symmetry using the parametric equations[more]
, , ,
An encoding of the knotted curve as an algebraic variety, (the intersection of two surfaces), defines a natural time parameter relative to the tangent geometry. It is then possible to integrate the period function by solving a second-order ordinary differential equation in the shape variable (see Details).[less]
There are many alternatives for and , all roughly equivalent. Here we have chosen
in order to make subsequent calculations slightly easier. Unfortunately, this definition introduces a spurious point at the origin. The extra point has no influence on the period integral, so we are safe to disregard it.
The time parameter must satisfy the consistency constraint
Using the vector notation
we calculate the time derivative as a cross product of gradient vectors,
This simply says that changes on a direction orthogonal to the local normal vectors of surfaces and . Equivalently, since solves and , the specified is a tangent vector to or . This is exactly what we need to change coordinates from . We would like to do so because the time coordinate is sensitive to singularities, and its period,
is a function of . The period integral is relatively easy to write, as is its annihilating operator,
𝒜=24 k (6-12 k2-k4+2 k6)&IndentingNewLine;-(1-k2)(16-102 k2+11 k4+15 k6)∂k&IndentingNewLine;-(1-k)2 k (1+k)2(2-k2) (4+k2)∂k2.
The annihilator can be proven correct using a certificate function for which
The source code contains the certificate and the relevant quality check. We omit details of the derivation, but will say that Hermite–Ostrogradsky reduction leads to the desired result very rapidly. In fact, the derivation algorithm we used is very similar to the one found in .
Once is known, we usually start analysis by characterizing its singular points, the roots of its final term. In this case, other than all singular points have magnitude at least 1. The transformation changes the chirality of the trefoil knot when or . When , the curve falls on the surface of a cylinder and crosses itself at three different four-way intersections. When , the curve reaches one six-way intersection at the origin. For any other , the curve unknots.
According to this analysis, we know all knot periods as long as we can solve the ODE over the domain . Specifically, chiral reflected knots are isoperiodic by symmetry, and must have . To solve the ODE, we need to calculate two integrals, say and , and then the two extra degrees of freedom are constrained. The rest of the work is done by NDSolve (as long as we can figure out how to set precision and accuracy goals).
The period function depends on choice of and . The convention given here leads to the simplest known annihilator . If is required not to include a spurious point at the origin, degrees of the -polynomial coefficients of will increase. We do not know of a convention where it is possible to make the annihilator of a rational function with no singularities in the domain .
This calculation is a spin-off from a discussion on the [math-fun] mailing list. The author would like to thank Cris Moore for suggesting that parametric equations could be programmatically transformed into algebraic varieties, Dan Asimov for offering critical comments and an alternative viewpoint, and many others for participating in an interesting discussion.
 B. Klee. "Approximating Pi with Trigonometric-Polynomial Integrals" from the Wolfram Demonstrations Project—A Wolfram Web Resource. (Jun 2, 2020) demonstrations.wolfram.com/ApproximatingPiWithTrigonometricPolynomialIntegrals.