https://www.dagstuhl.de/19352

### August 25 – 30 , 2019, Dagstuhl Seminar 19352

# Computation in Low-Dimensional Geometry and Topology

## Organizers

Maarten Löffler (Utrecht University, NL)

Anna Lubiw (University of Waterloo, CA)

Saul Schleimer (University of Warwick – Coventry, GB)

Erin Moriarty Wolf Chambers (St. Louis University, US)

## For support, please contact

## Documents

Dagstuhl Report, Volume 9, Issue 8

Aims & Scope

List of Participants

Dagstuhl's Impact: Documents available

Dagstuhl Seminar Schedule [pdf]

## Summary

One-dimensional structures embedded in higher-dimensional spaces are ubiquitous in both the natural and artificial worlds: examples include DNA strands, migration paths, planetary orbits, rocket trajectories, robot motion planning, chip design, and many more. These are studied in different areas of mathematics and computer science, under many names: knots, curves, paths, traces, trajectories, graphs, and others. However, researchers in many areas are just beginning to apply algorithmic techniques to find efficient algorithms for these structures, especially when more fundamental mathematical results are required. Broad examples of such problems include:

- classical algorithms on trajectories like the Fréchet distance as a way to formalize a distance measure as a curve changes;
- morphing between two versions of a common graph, which again tracks a higher dimensional space that corresponds to movement of a one-dimensional object;
- drawing and manipulating objects in three-manifolds, such as graphs, curves, or surfaces; and
- perhaps the simplest problem posed (in different ways) in all these areas, "how does one draw and morph a nice curve on a nice surface?"

This seminar was the second in a series. In the first seminar, the goal was to identify connections and seed new research collaborations along the spectrum from knot theory and topology, through to computational topology and computational geometry, and all the way to graph drawing. After the success of the first seminar, the goal for this second round was to continue and extend prior work, in particular by focussing on how objects change over time.

The seminar began with three overview talks from researchers in different areas (trajectory analysis, algorithmic topology, and graph drawing) to motivate and introduce problems which would fit the theme of changes over time in the representations of low-dimensional objects in higher dimensional spaces. We then invited all participants to describe open problems (most of which were circulated in advance of the meeting) that fit with the topic of the workshop and could benefit from broad expertise. For the remainder of the workshop we split into small working groups each focussed on a particular open problem.

Throughout the workshop we used Coauthor, a tool for collaboration designed by Erik Demaine (MIT), to share progress and updates among all the working groups. This, together with twice-daily progress reports, allowed us to share ideas and expertise among all participants, which was very effective. Another advantage was that we had a record of the work accomplished when the workshop ended.

Below, we (the organizers) describe the main working group topics and how they connected to the overarching theme. The abstracts of talks in the seminar and preliminary results from the working groups are also outlined later in this report.

One group worked on open questions that were motivated by 3-manifolds. In particular, they considered lower bounds for deciding the complexity of a knot or link equivalence, with a goal of finding specific knots that require many simplification moves. Their work involved both designing smaller examples, as well as doing larger scale exhaustive search using the software tool Regina.

Another group considered representations of graphs and hypergraphs by touching polygons in 3-d. They were able to leverage the dual graph of the polyhedral complex in 3d, and make progress on classifying which types of graphs could (or could not) be realized. Their problem was primarily combinatorial, but the techniques used included several interesting topological arguments about embeddings of manifolds into 3d or into 3-manifolds.

Several groups considered problems about flows or morphs of curves in various settings. One question centered on visualizing actual embedded homotopies on a given surface; there is considerable prior work on how to compute such homotopies between curves quickly, but it generally focuses on computing the complexity of the homotopy as opposed to the actual sequence of simplifications needed. The group looked more closely at this algorithm, and was able to outline a proof that in fact an extension of that algorithm would generate the actual homotopy, for a slightly higher time cost. A second group considered curves in the plane, and investigated options for computing a "nice" morph between them. As the question was more vague, the group did quite a bit of background investigation on prior work, and then discussed a new technique based on 3-manifolds and normal surface theory which might lead to a new family of morphs. A third group looked at the problem of preprocessing a given curve so that the Fréchet distance to any other query curve could be efficiently computed, and were able to obtain improved time bounds for several variants of the problem.

In summary, the workshop fostered a highly collaborative environment where combining the expertise and knowledge of researchers from different communities allowed us to solve problems of common interest across those communities. A major theme was how connected the various problems could be; often, a proof technique or piece of literature suggested by a member of a different community proved useful or insightful to a group working in a different domain.

**Summary text license**

Creative Commons BY 3.0 Unported license

Erin Moriarty Wolf Chambers, Maarten Löffler, Anna Lubiw, and Saul Schleimer

## Related Dagstuhl Seminar

## Classification

- Data Structures / Algorithms / Complexity

## Keywords

- Mathematics
- Topology
- Trajectories
- Graph drawing
- Knots