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P. Kindermann, S.G. Kobourov, M. Löffler, M. Nöllenburg, A. Schulz, B. Vogtenhuber:
"Lombardi Drawings Of Knots And Links";
Journal of Computational Geometry, 10 (2019), 1; 444 - 476.

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 (de ned 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 o set ", while maintaining a 180 angle between opposite edges.

http://dx.doi.org/10.20382/jocg.v10i1

https://publik.tuwien.ac.at/files/publik_284601.pdf