Mathematical knots are closed-loop curves embedded in Euclidean 3-space. I am turning such knots into geometrical sculpture models by sweeping a suitable cross-sectional profile along that curve, possibly letting the profile grow or shrink and twist as it moves along the knotted sweep curve.
“Cable-Knots” emerge when I use a “cable” with multiple parallel strands to follow the knot-curve. If every strand connects to itself at the point where the cable closes into a loop, one obtains several mutually interlinked mathematical knots. A more intriguing option is to give the cable some overall twist so that each strand connects with one of the other strands, thus forming a single strand corresponding to a more complex knot.

Artworks

The mathematical knot underlying this sculpture is the 3-crossing “Trefoil knot” balancing on one of its lobes and extending the other two lobes upwards. The “cable” that follows this knotted curve has three strands, each with a triangular cross-section itself. During one pass around the Trefoil loop, the cable twists through 120 degrees, so that strand A connects to strand B, strand B to C, and finally C to A. This results in just one single strand forming a much more complex mathematical knot. Wherever the original Trefoil knot had a local crossing, we now find a group of 3x3=9 crossings. In addition, two more crossings result from the Möbius-like twisting of the three strands. Thus, the final sculpture is a 29-crossing knot!

A “cable knot” can also be understood as a “warped torus knot,” where the underlying donut around which the knotted strand has been wound is itself deformed into a mathematical knot. In the present sculpture, the underlying knot is the 4-crossing “Figure-8 knot” standing on two of its lobes and extending the other two lobes upwards. The “cable” following this knotted curve has three strands, each with a triangular cross-section itself. During one pass around the Figure-8 loop, the cable twists through 120 degrees, and this Möbius-like twisting generates a final sculpture that is a single mathematical knot with 4 x (3x3) + 2 = 38 crossings.