I am using mathematical knots as an inspiration for making abstract geometrical sculptures. Recently, I have started with simple knots and have then increased their complexity by following the knot curve with a “cable” with several parallel strands. Before the cable is closed into a loop, it is given an appropriate amount of torsional twist, so that we obtain cyclic connectivity among all the strands, resulting in a single closed-loop strand and thus in a valid mathematical knot.
On the left is the 4-crossing “Figure-8 knot” outlined with a 4-strand cable. The cable is given a 90° torsional twist to connect the four strands into a single closed loop. Wherever the Figure-8 knot had a simple crossing, there are now 16 crossings; and there are three more crossings due to the quarter twist of the cable. Thus, the resulting knot has 67 crossings.
On the right is "Knot-5_2" outlined with a 4-strand cable. This leads to a knot with 5×16=80 crossings. In addition, the four strands are woven into a classical braid, adding another 129 crossings, since there are 10.75 braid periods, and each period has 12 crossings. This leads to a knot with a grand total of 209 crossings.