https://0x0.st/ifEP.txt https://0x0.st/ifEX.txt Meta-Data, by (line #): (1) (p,q) torus knot Parametric Eqs, (2) Z-linear surface, degree q, (3) Z-quadratic surface, degree 4, (4) dt/d\phi, (5) Annihilator of dt/du, an ODE in k, (6) Annihilator Certificate, A[dt/du]-d/dt(cert)=0. Imposing Symmetry leaves one degree of freedom. After trying many different constraints, I found that allowing, in fact requiring, an extra point (X,Y,Z)=(0,0,0) leads to smallest known proof data, see linked certificates above. For the first two cases (p,q) the Z-linear surface had degree q, and the ODE has q terms. This suggests apparent genus 1 and 2 for the first two torus knots, (2,3) and (3,4). When measuring genus by period ODE, it often happens that apparent genus is less than real genus due to hidden symmetry. I don't know about slice genus, but according to the proved Milnor conjecture, the numbers are 1 and 3, see corollary 1 of: https://arxiv.org/abs/math/0402131 I doubt that any representation of the stevedore knot would have a rational period function, but it is Lissajous, so it might possible to check. Unfortunately, I don't have anymore time this week. --Brad