2024 Joint Mathematics Meetings

# Carlo H. Séquin

## Artists

## Statement

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 and shrink and twist as it moves along the knotted sweep curve. A quick way to increase the complexity of the knot is to follow the knot-curve with a “cable” comprised of multiple parallel strands. This set of cable strands can be connected into a single mathematical knot, if the cable is given some overall twist, so that each strand connects with one of the other strands in a cyclic manner. Knot complexity can be further enhanced by weaving the strands into a braid.

## Artworks

This sculpture started out as a triply-twisted Möbius band forming a Trefoil knot. This band was then replaced by a flat 4-strand braid. The length of this braid was carefully tailored so that the desired cyclic connectivity between the four strands was obtained. As a simple 4-strand cable knot, it would already exhibit 3*4*4 +3 = 51 crossings. Using 12+1/4 periods of a 4-strand braid, where each period introduces an additional 12 crossings, then results in a braided Trefoil knot with 198 crossings.

Next, I apply the braiding operation in a recursive manner to a somewhat simpler structure. I start with a 4-strand braid forming an annulus. 2+1/4 periods of this braid result in a mathematical knot with 27 crossings. Turning this “Annulus Braid” into a 3-strand cable knot, multiplies its 27 crossings by a factor of 9. A secondary braiding operation, creating 53+1/3 periods along the original knot strand, adds another 320 crossings, for a grand total of 563 crossings!
The four 3-strand braid segments joining to form this knot are shown with twelve different colors to make the recursive braiding more visible and easier to understand.