Lombardi drawings of knots and links

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR², such that no more than two points project to the same point in lR². These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR³, so their projections should be smooth curves in lR² with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defned by circular-arc edges and perfect angular resolution). We show that several knots do not allow crossing-minimal plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have plane Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a crossing-minimal 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 a plane Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180^◦ angle between opposite edges.