Hi Dan, The complex period of the knot is determined by its winding numbers multiplied by the real and complex periods of the Riemann surface it is drawn upon, then added together. The Riemann surface I used is a complex extension of an elliptic curve in Harold Edwards’s normal form, see also: https://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01153-6/ It is a torus and can be drawn parametrically in four dimensions using elliptic functions such as defined by Edwards. To do so, you need to calculate real and complex periods. When z is chosen real between 0 and 1, the periods in Edwards’s model are TR=4*K(z) and TC=i*2*K(1-z). (The factor of 2 comes from a difference of harmonic frequencies around hyperbolic points.) The elliptic functions are defined over the complex plane but they are doubly periodic, with periods TR and TC. Torus knots (n,m) can be drawn by taking non-crossing paths through the complex plane that close modulo the period lattice. The trajectory through the complex plane determines a trajectory through C^2, which we can then map somehow into R^3. (This part is not easy, but doesn’t affect the period integral.) To calculate the period, we simply count n real divisions horizontal and m complex divisions vertical, until we land on the same location in the period rectangle. The period is then T=n*TR+m*TC, and that is the amount of complex time it takes to get once around the curve. In this first example I choose z=1/2 so that we get a period square, and n=3, m=4 to accentuate the dihedral symmetry. To reiterate, the period is: T=(3*4+4*2*i)*K(1/2). On the same torus, other periods would be something like T=(n*4+m*2*i)*K(1/2). On other similar tori, we would have to write T=(n*4*K(z)+m*2*i*K(1-z)). Another example drawing, z=1/4: https://0x0.st/ip1O.png (looks slightly better for low z, maybe) For the trefoil knot it would be more natural to use a cubic elliptic curve with triangular dihedral symmetry. There is a choice of coordinates where TR=K3(z) and TC=(sqrt(3)/3)*K3(1-z), with K3(z)=2*Pi*2F1(1/3,2/3,1,z), so we can calculate the period of a (n,m) knot, on any surface indexed by z: T = 2*Pi*n*2F1(1/3,2/3,1,z) + 2*Pi*(sqrt(3)/3)*2F1(1/3,2/3,1,1-z). It would be awesome to see a symmetry-preserving drawing of the (2,3) knot on an elliptic curve, but not so easy, due to the difficulty of the following problem: Given a form for elliptic curves: z = 2H= q^2 + p^2 - sqrt(4/27)*(3*q*p^2 - q^3) find the parameterizing elliptic function p(t) and q(t). (Also Is z=2H a normal form for elliptic curves?) --Brad On May 24, 2020, at 12:43 PM, Dan Asimov <dasimov@earthlink.net> wrote: Brad, I don't yet understand your definition of the period of a knot. Could you please tell me (or math-fun) what that definition is? (I did peek at your nicely illustrated paper on github, but did not readily extract such a definition.) Thanks, Dan