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P. Kindermann, S.G. Kobourov, M. Löffler, M. Nöllenburg, A. Schulz, B. Vogtenhuber:
"Lombardi Drawings of Knots and Links";
Vortrag: International Symposium on Graph Drawing and Network Visualization (GD), Boston; 25.09.2017 - 27.09.2017; in: "Graph Drawing and Network Visualization (GD 2017)", F. Frati, K.-L. Ma (Hrg.); Springer Lecture Notes in Computer Science, 10692 (2018), ISBN: 978-3-319-73914-4; S. 113 - 126.

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 continu- ity 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.

graph drawing, knot theory, knot diagrams

http://dx.doi.org/10.1007/978-3-319-73915-1_10

https://arxiv.org/abs/1708.09819v1