Publications in Scientific Journals:
P. Kindermann, S.G. Kobourov, M. Löffler, M. Nöllenburg, A. Schulz, B. Vogtenhuber:
"Lombardi Drawings Of Knots And Links";
Journal of Computational Geometry,
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.
"Official" electronic version of the publication (accessed through its Digital Object Identifier - DOI)
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.