# Lombardi Drawings of Knots and Links

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into R^2, such that no more than two points project to the same point in R^2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in R^3, so their projections should be smooth curves in R^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 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.

• 25 publications
• 51 publications
• 42 publications
• 51 publications
• 9 publications
• 28 publications
08/18/2019

### Graphs with large total angular resolution

The total angular resolution of a straight-line drawing is the minimum a...
12/04/2008

### Stroke Fragmentation based on Geometry Features and HMM

Stroke fragmentation is one of the key steps in pen-based interaction. I...
06/05/2023

### On the Complexity of Lombardi Graph Drawing

In a Lombardi drawing of a graph the vertices are drawn as points and th...
07/17/2019

### An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants

A drawing of a graph in the plane is pseudolinear if the edges of the d...
04/11/2017

### Reconstruction of 3-D Rigid Smooth Curves Moving Free when Two Traceable Points Only are Available

This paper extends previous research in that sense that for orthogonal p...
09/27/2021

### The edge labeling of higher order Voronoi diagrams

We present an edge labeling of order-k Voronoi diagrams, V_k(S), of poin...
11/09/2021

### Dynamics of charged particles moving around Kerr black hole with inductive charge and external magnetic field

We mainly focus on the effects of small changes of parameters on the dyn...