Joel Hass - Research Summary

Somewhat Current CV (PDF file)

My research centers on low-dimensional topology and geometry.

Publications by Date (PDF file)

Publications by Area (PDF file)

Recent Papers and Preprints


Approximating Surfaces in R3 by Meshes with Guaranteed Regularity.
(with M. Trnkova)
Also at .
Python code available at gitlab.
We study the problem of approximating a surface by a high quality mesh. The GradNormal algorithm gives a piecewise-differentiable approximation with angles in the interval [35.2, 101.5] (degrees) as the mesh becomes increasingly fine. Previous angle bounds were in the interval [30, 120] (degrees).


Tangle decompositions of alternating link complements, at arxiv.
(with A Thompson and A. Tsietkova)
We refine results of Menasco and Thistlethwaite to show that if a tangle decomposition exists in an alternating link diagram, then either it is visible or the link is of a particular form, which we call a pseudo-Montesinos link.


Universal Knot Diagrams at arxiv, and Universal Knot Diagrams at JKTR.
(with C. Even-Zohar, N. Linial, T. Nowik)
We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.

Computing trisections of 4-manifolds.
(with Mark Bell, J. Hyam Rubinstein, Stephan Tillmann)
Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a 4-dimensional manifold as a union of three 4-dimensional handlebodies. The complexity of the 4-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or 4-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.

Alternating links have at most polynomially many Seifert surfaces of fixed genus.
(with A Thompson and A. Tsietkova)
Let L be a non-split prime alternating link with n>0 crossings. We show that for each fixed g, the number of genus-g Seifert surfaces for L is bounded by an explicitly given polynomial in n. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponential.


The Distribution of Knots in the Petaluma Model.
(with C. Even-Zohar, N. Linial, T. Nowik)
The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of every specific knot decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots.


Isoperimetric Regions in Nonpositively Curved Manifolds.
We show that isoperimetric regions in two and three-dimensional nonpositively curved manifolds are not necessarily balls, and need not even be connected.


Minimal Fibrations of Hyperbolic 3-manifolds.
We give examples of hyperbolic 3-manifolds that fiber over the circle but do not admit minimal fibrations.

Landmark-free geometric methods in biological shape analysis.
(with P. Koehl)
We propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates.

A Metric for genus-zero surfaces.
(with P. Koehl)
We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms give a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.

The number of surfaces of fixed genus in an alternating link complement.
(with A. Thompson and A. Tsvietkova)
Let L be a prime alternating link with n crossings. We show that for each fixed g, the number of genus g incompressible surfaces in the complement of L is bounded by a polynomial in n. Previous bounds were exponential in n.


Invariants of Random Knots and Links.
(with C. Even-Zohar, N. Linial, T. Nowik)
We study random knots and links in using the Petaluma model. We obtain formulae for the distribution of the linking number of a random two-component link and for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3.

How round is a protein? Exploring protein structures for globularity using conformal mapping.
(with P. Koehl)
We present a new approach to measuring the roundness of a genus-zero surface. We then test and apply this to measure the roundness of the platonic solids and the roundness of a collection of 533 proteins.

Automatic Alignment of Genus-Zero Surfaces.
(with P. Koehl)
Gives a new method to automatically find optimal conformal mappings between two triangulated spherical surfaces.

Simplical Energy and Simplicial Harmonic Maps
(with P. Scott)
This paper defines a new type of energy function on maps from a triangulated surface to a manifold. The energy captures many proerties of the Dirchlet energy, but it is much simpler to establish properties such as existence, uniqueness etc.

Topological And Physical Link Theory Are Distinct
(with A. Coward)
We show that the theory of classical knots and links does not coincide with that of knots and links having thickness and fixed length, as in the real world. In particular we prove the existence of a two component link that is split in the classical theory but cannot be split with a physical isotopy.


Globally Optimal Cortical Surface Matching With Exact Landmark Correspondence.
(with A. Tsui, D. Fenton, P. Vuong, P. Koehl, N. Amenta, D. Coeurjolly, C. DeCarli, and O. Carmichael)
A paper from the Information Processing in Medical Imaging conference. This applies hyperbolic orbifolds to align brain cortex surfaces with prescribed landmarks.

What is an Almost Normal Surface?
We explain how almost normal surfaces emerge naturally from the study of geodesics and minimal surfaces.

Many of my recent preprints are available on the math arxives. and on

Research Areas

Three dimensional manifolds
Computational complexity in topology
Minimal surfaces and bubbles
Geometric problems in Biology

The focus of my research is in three dimensional geometry and topology. This subject contains many fascinating problems, with important applications.

Geometric Design

I worked on problems related to Computer Aided Design with Rida Farouki. This work can be seen here.

Minimal Surfaces

Minimal Surfaces and Constant Mean Curvature Surfaces are models for soap films and for optimizing shapes that enclose given volumes.

Click here for information and graphics on Double Bubbles.

Curve Evolution

I have looked at various aspects of the motions of curves and surfaces, with dynamics related to the curvature.

Click here for material on Curve Flows. You can draw a curve and see it flow on your web browser. (Added: unfortunately security updates to java have broken this code.)


In work with Jeff Lagarias, Nick Pippenger, Ian Agol and Bill Thurston, I studied the computational complexity of problems associated with unknotting. Tahl Nowik and I developed invariants of knot diagrams which gave quadratic lower bounds on the number of Reidemeister moves needed to connect distinct diagrams of a knot.

Geometric Biology

I am working with Patrice Koehl on geometric problems arising in biology.

The behavior of proteins and other biological molecules is significantly affected by their geometry. Many fascinating geometric problems are closely related to biological applications.

We are working with Nina Amenta, and Owen Carmichael on a project in shape analysis and shape matching. These problems connect to brain imaging and the cortical surface of the brain. It also connects to problens such as facial recognition and understanding the relations between fossils and evolutionary trees.

My Recent Conferences and trips:

Symposium on Geometric Processing, SGP2020
Utrecht, NL (virtual)
July 6-8, 20208.

Institut Henri Poincare
School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects
June 18-22, 2018.

Computational Topology and Geometry workshop.
July 10-19, 2017.

My 60th birthday conference:
Geometry, Topology and Complexity of Manifolds, and applications to Biology
UC Berkeley
May 20-22, 2015.

Geometric Structures on 3-manifolds
School of Mathematics, Institute for Advanced Study
September 1, 2015 to April 30, 2016
I was on sabbatical in 2015-2016, at the above program.

Applied Topology and High-Dimensional Data Analysis
University of Victoria, British Columbia, Canada
August 17-28, 2015

Congreso Matematica Capricornio
Iquique - Chile
August 5-7, 2015

AMS Sectional Meeting, Las Vegas
Spring Western Sectional Meeting
University of Nevada, Las Vegas, Las Vegas, NV
April 18-19, 2015

In and Around Combinatorics
The 18th Midrasha Mathematicae
Israel Institute for Advanced Studies
January 18-30, 2015

Hyperbolic Geometry and Minimal Surfaces
IMPA, Rio de Janeiro
01/04/15 to 01/10/15

Computational Topology and Geometry workshop.
Dec. 15-17, 2014.