Knot Divided (January 2005)

Dan Schwalbe, Stan Wagon, Richard Seeley, John Sullivan (and Carlo Séquin, designer)

Carlo wrote: Can a DIVIDED KNOT be NOT DIVIDED? We start with the simplest possible knot—the overhand knot, also known as the trefoil or pretzel knot—which we then split lengthwise along the whole strand that forms the three loops. But there is a twist that may lead to surprises: The knotted strand is actually a triply twisted Moebius band!

Thus the question:  Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand?

FOR MATHEMATICIANS ONLY: There is a self-referential beauty in our sculpture: If one forms a Möbius band by twisting a belt through three half-turns (instead of just one), then the band’s edge forms a trefoil knot. Mathematicians classify the complexity of knots by the minimum number of line crossings that one must have when trying to draw that knot on a piece of paper. 

The trefoil knot is the simplest knot and cannot be drawn with fewer than three crossings. Our sculpture starts out as a triply twisted Möbius band, knotted into a trefoil knot. When we split it lengthwise, it is still a single knot, but of higher complexity. Can you figure out what its crossing number is?