About

Nathan M. Dunfield's research area is the topology and geometry of 3-manifolds. He was drawn to this field because of the richness introduced by Thurston's revolutionary work starting in the 1970s, particularly Thurston's insight that many 3-manifolds admit homogeneous Riemannian metrics and that the topology of a 3-manifold can be studied through its geometry. This geometric perspective has been stunningly confirmed by Perelman's proof of the Geometrization Conjecture. Although Dunfield's work initially focused on purely topological problems, he has employed a broad range of techniques including hyperbolic geometry, number theory, algebraic geometry, combinatorial group theory, and the theory of foliations. These interdisciplinary connections have led him to collaborate with number theorists, theoretical physicists, and computer scientists. His research has incorporated advanced concepts such as the Langlands Conjecture and the Classification of Finite Simple Groups, as well as explorations of topological phenomena like "random 3-manifolds." Dunfield has been a faculty member at the University of Illinois Urbana-Champaign since 2007. Prior to that, he spent four years at Harvard and four years at Caltech after earning his PhD from the University of Chicago in the late 20th century. In recognition of his contributions, he became a Fellow of the American Mathematical Society in 2013.

Research topics

• Genetics
• Pure mathematics
• Philosophy
• Mathematics
• Epistemology
• Biology

Selected publications

• Ribbon concordances and slice obstructions: experiments and examples

  arXiv (Cornell University) · 2025-12-26

  preprintOpen access1st authorCorresponding

  There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in either setting, we are able to determine it smoothly for all but about 11,400 knots (0.003% or 1 in 30,000) and topologically for all but about 1,400 knots (0.0004% or about 1 in 250,000). In particular, we show that some 1.6 million of these knots (0.46%) are smoothly slice (in fact ribbon) and that 350.5 million are not even topologically slice (99.54%). We use a wide range of tools and techniques, and introduce several new or refined methods for probing these properties. Along the way, we produce 500,000 pairs of 0-friends, that is, pairs of distinct knots with the same 0-surgery. We discuss how our data is consistent with several important conjectures and suggests new ones, and highlight the simplest knots where sliceness remains unknown.

  PublisherDOI

• Roots of Alexander polynomials of random positive 3-braids

  Advances in Mathematics · 2025-04-09

  articleOpen access1st authorCorresponding

  Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.

  PublisherDOI

• Ribbon concordances and slice obstructions: code and data

  Harvard Dataverse · 2025-12-22

  datasetOpen access1st authorCorresponding

  Code and data to accompany the paper <i>Ribbon concordances and slice obstructions: experiments and examples</i> by Nathan Dunfield and Sherry Gong.

  PublisherDOI

• Ribbon concordances and slice obstructions: experiments and examples

  ArXiv.org · 2025-12-26

  articleOpen access1st authorCorresponding

  There are 352.2 million prime knots in the 3-sphere with at most 19 crossings. We study which of these knots are slice, in both the smooth and topological categories. While no algorithm is known for deciding whether a given knot is slice in either setting, we are able to determine it smoothly for all but about 11,400 knots (0.003% or 1 in 30,000) and topologically for all but about 1,400 knots (0.0004% or about 1 in 250,000). In particular, we show that some 1.6 million of these knots (0.46%) are smoothly slice (in fact ribbon) and that 350.5 million are not even topologically slice (99.54%). We use a wide range of tools and techniques, and introduce several new or refined methods for probing these properties. Along the way, we produce 500,000 pairs of 0-friends, that is, pairs of distinct knots with the same 0-surgery. We discuss how our data is consistent with several important conjectures and suggests new ones, and highlight the simplest knots where sliceness remains unknown.

  PublisherOA PDF

• A unified Casson–Lin invariant for the realforms of SL(2)

  Geometry & Topology · 2025-11-26 · 2 citations

  articleOpen access1st authorCorresponding

  We introduce a unified framework for counting representations of knot groups into SU 2 and SL 2 R.For a knot K in the 3-sphere, Lin and others showed that a Casson-style count of SU 2 representations with fixed meridional holonomy recovers the signature function of K.For knots whose complement contains no closed essential surface, we show there is an analogous count for SL 2 R representations.We then prove the SL 2 R count is determined by the SU 2 count and a single integer h.K/, allowing us to show the existence of various SL 2 R representations using only elementary topological hypotheses.Combined with the translation extension locus of Culler and Dunfield, we use this to prove left-orderability of many 3-manifold groups obtained by cyclic branched covers and Dehn fillings on broad classes of knots.We give further applications to the existence of real parabolic representations, including a generalization of the Riley conjecture (proved by Gordon) to alternating knots.These invariants exhibit some intriguing patterns that deserve explanation, and we include many open questions.The close connection between SU 2 and SL 2 R comes from viewing their representations as the real points of the appropriate SL 2 C character variety.While such real loci are typically highly singular at the reducible characters that are common to both SU 2 and SL 2 R, in the relevant situations we show how to resolve these real algebraic sets into smooth manifolds.We construct these resolutions using the geometric transition S 2 !E 2 !H 2 , studied from the perspective of projective geometry, and they allow us to pass between Casson-Lin counts of SU 2 and SL 2 R representations unimpeded.

  PublisherOA PDFDOI

• Roots of Alexander polynomials of random positive 3-braids

  arXiv (Cornell University) · 2024-02-09

  preprintOpen access1st authorCorresponding

  Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.

  PublisherOA PDFDOI

• Enumeration and Identification of Unique 3D Spatial Topologies of Interconnected Engineering Systems Using Spatial Graphs

  Journal of Mechanical Design · 2023-07-18 · 4 citations

  article

  Abstract Systematic enumeration and identification of unique 3D spatial topologies (STs) of complex engineering systems (such as automotive cooling systems, electric power trains, satellites, and aero-engines) are essential to navigation of these expansive design spaces with the goal of identifying new spatial configurations that can satisfy challenging system requirements. However, efficient navigation through discrete 3D ST options is a very challenging problem due to its combinatorial nature and can quickly exceed human cognitive abilities at even moderate complexity levels. This article presents a new, efficient, and scalable design framework that leverages mathematical spatial graph theory to represent, enumerate, and identify distinctive 3D topological classes for a generic 3D engineering system, given its system architecture (SA)—its components and their interconnections. First, spatial graph diagrams (SGDs) are generated for a given SA from zero to a specified maximum number of interconnect crossings. Then, corresponding Yamada polynomials for all the planar SGDs are generated. SGDs are categorized into topological classes, each of which shares a unique Yamada polynomial. Finally, within each topological class, 3D geometric models are generated using the SGDs having different numbers of interconnect crossings. Selected case studies are presented to illustrate the different features of our proposed framework, including an industrial engineering design application: ST enumeration of a 3D automotive fuel cell cooling system (AFCS). Design guidelines are also provided for practicing engineers to aid the application of this framework to different types of real-world problems such as configuration design and spatial packaging optimization.

  PublisherDOI

• On the rank of knot homology theories and concordance

  arXiv (Cornell University) · 2023-03-07

  preprintOpen access1st authorCorresponding

  For a ribbon knot, it is a folk conjecture that the rank of its knot Floer homology must be 1 modulo 8, and another folk conjecture says the same about reduced Khovanov homology. We give the first counter-examples to both of these folk conjectures, but at the same time present compelling evidence for new conjectures that either of these homologies must have rank congruent to 1 modulo 4 for any ribbon knot. We prove that each revised conjecture is equivalent to showing that taking the rank of the homology modulo 4 gives a homomorphism of the knot concordance group. We check the revised conjectures for 2.4 million ribbon knots, and also prove they hold for ribbon knots with fusion number 1.

  PublisherOA PDFDOI

• Computing a Link Diagram From Its Exterior

  Discrete & Computational Geometry · 2023-08-02 · 1 citations

  articleOpen access1st author

  A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

  PublisherOA PDFDOI

• Local equivalence and refinements of Rasmussen's s-invariant

  arXiv (Cornell University) · 2023-12-14

  preprintOpen access1st authorCorresponding

  Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even-odd (LEO) triple. We get a homomorphism from the smooth concordance group $C$ to the resulting local equivalence group $C_{LEO}$ of such triples. We give several versions of the $s$-invariant that descend to $C_{LEO}$, including one that completely determines whether the image of a knot $K$ in $C_{LEO}$ is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu-Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.

  PublisherOA PDFDOI

Recent grants

• Facets of low-dimensional topology

  NSF · $387k · 2015–2018

• Surfaces in finite covers of 3-manifolds and aspects of the mapping class groups

  NSF · $280k · 2007–2011

• Facets of the topology and geometry of 3-manifolds

  NSF · $185k · 2011–2014

• Facets of the Topology and Geometry of 3-Manifolds

  NSF · $367k · 2018–2023

Frequent coauthors

• Danny Calegari

  8 shared

• Stavros Garoufalidis

  Southern University of Science and Technology

  8 shared

• Kai A. James

  Georgia Institute of Technology

  5 shared

• Lawrence Zeidner

  RTX (United States)

  4 shared

• Stefan Friedl

  University of Regensburg

  4 shared

• Dinakar Ramakrishnan

  4 shared

• Ian Agol

  University of California, Berkeley

  4 shared

• Jean Raimbault

  Château Gombert

  4 shared

Labs

• Nathan M. Dunfield's LabPI

  Topology and geometry of 3-manifolds

Awards & honors

• Fellow of the American Mathematical Society (2013)