Kindermann, P., Kobourov, S., Löffler, M., Nöllenburg, M., Schulz, A., & Vogtenhuber, B. (2018). Lombardi Drawings of Knots and Links. In F. Frati & K.-L. Ma (Eds.), Graph Drawing and Network Visualization. GD 2018 (pp. 113–126). Springer. https://doi.org/10.1007/978-3-319-73915-1_10
26th International Symposium on Graph Drawing and Network Visualization (GD 2018)
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Event date:
26-Sep-2018 - 28-Sep-2018
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Event place:
Barcelona, Spain
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Number of Pages:
14
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Publisher:
Springer, Cham
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Peer reviewed:
Yes
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Keywords:
graph drawing; knot theory; knot diagrams
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Abstract:
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.
