I have had a longstanding fascination with knots, links, and orderly tangles. I have also been interested in smooth, closed surfaces of different genera, optimized to minimize some energy functional; these surfaces include the bending-energy-minimizing Lawson surfaces, the minimum-variation surfaces, as well as the "Optiverse," a minimum-energy half-way state of Morin's sphere-eversion process. In my recent work, these domains have started to intersect. I am looking for many topologically different, aesthetically pleasing ways to embed knots and links in smooth handle-bodies. My submission to this year's art exhibit complements my presentation on "Everted Embeddings."
A composition of four panels illustrates four symmetrical, but topologically different embeddings of the three Borromean rings on handle-bodies of genus 3. On the left, I start with the most symmetrical embedding possible on a Tetrus body, where two different loops wind around each of the six tetrahedral handles. Below, in the D3-symmetrical embedding, the red, green, and blue loops can wind around the three handles an arbitrary number of times without changing the topology of the Borromean link. These two embeddings are then everted by pulling the host surface through a small puncture placed in a suitable location. This results in two new, topologically different embeddings shown on the right.