Lombardi Drawings of Knots and Links

TL;DR

This paper explores Lombardi drawings of knots and links, showing limitations for certain knots, identifying classes that admit such drawings, and proposing relaxations that allow all knots to be represented with near-Lombardi or 2-Lombardi drawings.

Contribution

It demonstrates the limitations of Lombardi drawings for some knots and introduces relaxations that enable all knots to be represented with near-Lombardi or 2-Lombardi drawings.

Findings

Some knots do not admit plane Lombardi drawings.

A large class of 4-regular plane multigraphs have Lombardi drawings.

Every knot admits a plane 2-Lombardi drawing and is near-Lombardi with relaxed angular resolution.

Abstract

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in $R^3$, so their projections should be smooth curves in $R^2$ 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 $\varepsilon$, while maintaining a $180^\circ$ angle between opposite edges.