Lombardi Drawings of Knots and Links

Kindermann, Philipp and Kobourov, Stephen G. and Löffler, Maarten and Nöllenburg, Martin and Schulz, André and Vogtenhuber, Birgit (2017) Lombardi Drawings of Knots and Links. In: Graph Drawing and Network Visualization, GD 2017, September 25-27 , pp. 113-126(Official URL: https://doi.org/10.1007/978-3-319-73915-1_10).

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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 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 ε, while maintaining a 180∘ angle between opposite edges.

Item Type: Conference Paper
Classifications: G Algorithms and Complexity > G.560 Geometry
P Styles > P.999 Others
URI: http://gdea.informatik.uni-koeln.de/id/eprint/1594

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