Lombardi drawings of knots and links

Philipp Kindermann
Stephen G. Kobourov
Maarten Löffler
Martin Nöllenburg
André Schulz
Birgit Vogtenhuber
Artikel in Zeitschriften
erschienen in:
Journal of Computational Geometry, Volume 10, No 1, pp 444-476, 2019

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into IR^2, such that no more than two points project to the same point in IR^2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in IR^3, so their projections should be smooth curves in IR^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 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 offset ε, while maintaining a 180° angle between opposite edges.

@article{DBLP:journals/jocg/KindermannKLN0V19, author = {Philipp Kindermann and Stephen G. Kobourov and Maarten L{\"{o}}ffler and Martin N{\"{o}}llenburg and Andr{\'{e}} Schulz and Birgit Vogtenhuber}, title = {Lombardi drawings of knots and links}, journal = {J. Comput. Geom.}, volume = {10}, number = {1}, pages = {444--476}, year = {2019}, url = {}, doi = {10.20382/jocg.v10i1a15}, timestamp = {Thu, 10 Sep 2020 13:17:54 +0200}, biburl = {}, bibsource = {dblp computer science bibliography,} }
Christoph Doppelbauer | 10.05.2024