SQUARE KNOT, GRANNY KNOT  Two versions of the square knot.

 Traditional knots. Other names of the square knot: reef knot; other names of the granny knot: pigtail knot. Wikipedia: reef knot, granny knot. Images made by Alain Esculier.

 View from above: (hypotrochoid with 4 branches in which the first circular motion was replaced by an elliptic one). Maple program for the view in 3d of the square knot: e:=0.01: x, y, z := 3*sin(t)+2*sin(3*t), cos(t)-2*cos(3*t), 0.2*(cos(t)+2*sin(4*t)): display(  tubeplot({[-x, y, z],[x, y, z]}, t = Pi-e .. 2*Pi+e, radius = .15), orientation = [90, 0], grid = [300, 20], style = patchnogrid, lightmodel =light2, axes=none, scaling=constrained);  The open square (on the left) and granny (on the right) knots are the knots with 6 crossings represented opposite.  If the free blades with different colors are connected, we get the mathematical square and granny knots.  The first one is the sum of two trefoil knots mirror images of one another, the second is the sum of two trefoil knots that have the same orientation. They are the composite knots with the minimal number of crossings.  © Robert FERRÉOL  2018