Lombardi Drawings of Knots and Links

摘要

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into IR~2, such that no more than two points project to the same point in IR~2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in IR~3, so their projections should be smooth curves in IR~2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180° angle between opposite edges.