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arXiv · 1708.09819

Lombardi Drawings of Knots and Links

Philipp Kindermann·Stephen Kobourov·Maarten Löffler·Martin Nöllenburg·André Schulz·Birgit Vogtenhuber
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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.

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Philipp Kindermann, Stephen Kobourov, Maarten Löffler, Martin Nöllenburg, André Schulz, Birgit Vogtenhuber. 2019-03-11. Lombardi Drawings of Knots and Links. https://arxiv.org/abs/1708.09819

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