K_{3,3} embedded on a torus

utility.jscad

/*jslint node*/
"use strict";
const jscad = require("@jscad/modeling");
const {vec2, vec3} = jscad.maths;
const {colorize} = jscad.colors;
const {cylinder, sphere, torus} = jscad.primitives;
const {rotate, scale, translate} = jscad.transforms;

let edgeCylinder = cylinder({height: 1, radius: 0.01});  // cached

function edge(v, w) {
    let d = vec3.subtract(vec3.create(), w, v);
    let m = vec3.lerp(vec3.create(), v, w, 0.5);
    let R = [0, Math.acos(d[2] / vec3.length(d)), Math.atan2(d[1], d[0])];
    let S = [1, 1, vec3.length(d)];
    return translate(m, rotate(R, scale(S, edgeCylinder)));
}

function vertex(c) {
    return translate(c, sphere({radius: 0.08}));
}

function xyz(a) {
    let M = [0, Math.sin(a[1]) / 2 - 1, Math.cos(a[1]) / 2];
    return vec3.rotateZ(vec3.create(), M, [0, 0, 0], a[0]);
}

function torus_edge(v, w) {
    const n = 100;
    let ret = [];
    let from = v;
    let i = 1;
    while (i <= n) {
        let to = vec2.lerp(vec2.create(), v, w, i / n);
        ret.push(edge(xyz(from), xyz(to)));
        from = to;
        i += 1;
    }
    return ret;
}

function main() {
    let Vs = [
        [-Math.PI / 5, -1.2 * Math.PI / 4],
        [-Math.PI / 5, -0.2 * Math.PI / 4],
        [0, -1.2 * Math.PI / 4],
        [0, -0.2 * Math.PI / 4],
        [Math.PI / 5, -1.2 * Math.PI / 4],
        [Math.PI / 5, -0.2 * Math.PI / 4],

        [2 * Math.PI - Math.PI / 5, -1.2 * Math.PI / 4],              // 0
        [0, 2 * Math.PI - 1.2 * Math.PI / 4],                         // 2
        [Math.PI / 5 - 2 * Math.PI, 2 * Math.PI - 1.2 * Math.PI / 4]  // 4
    ];
    let Es = [[7, 1], [2, 3], [7, 5], [0, 1], [0, 3], [4, 3], [4, 5], [6, 5]];
    Es.push([8, 1]);

    let ret = [
        colorize([0.66, 0.0, 0.66], torus({innerRadius: 0.5, outerRadius: 1}))
    ];
    Vs.forEach(function (v) {
        ret.push(colorize([1, 0.5, 0], vertex(xyz(v))));
    });
    Es.forEach(function (e) {
        ret.push(colorize([1, 1, 0], torus_edge(Vs[e[0]], Vs[e[1]])));
    });
    return ret;
}

module.exports = {main};

Same embedding on Möbius strip here:
https://gist.github.com/Hermann-SW/3950cd4da9f955a4562066b74a0f573c?permalink_comment_id=5590529#gistcomment-5590529