Research

Exotic aspherical 4-manifolds
with Michael Davis, Jingyin Huang, Daniel Ruberman, and Nathan Sunukjian
Preprint (2024)

We construct closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic. These provide counterexamples to a smooth analog of the Borel conjecture in dimension four.  Our technique is to apply the "reflection group trick" of the first author to pairs of exotic 4-manifolds with boundary constructed by the second author and Piccirillo.  (arxiv / talk)

An atomic approach to Wall-type stabilization problems
Preprint (2023)

Wall-type stabilization problems investigate the collapse of exotic 4-dimensional phenomena under stabilization operations (e.g., taking connected sums with S²xS²). We propose an elementary approach to these problems, providing a construction of exotic 4-manifolds and knotted surfaces that are candidates to remain exotic after stabilization — including examples in the setting of closed, simply connected 4-manifolds. As a proof of concept, we show this construction yields exotic surfaces in the 4-ball that remain exotic after (internal) stabilization, detected by the cobordism maps on universal Khovanov homology. 

We also compare these Khovanov-theoretic obstructions for surfaces to the Floer-theoretic counterparts for exotic 4-manifolds obtained as their branched covers, suggesting a bridge via Lin’s spectral sequence from Bar-Natan homology to involutive monopole Floer homology. (arxiv)

Seifert surfaces in the 4-ball
with Seungwon Kim, Maggie Miller, JungHwan Park, and Isaac Sundberg
Journal of the European Mathematical Society  (to appear)

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in the 3-sphere that do not become isotopic when their interiors are pushed into the 4-ball . We give examples where the surfaces are not topologically isotopic in the 4-ball, as well as examples that are topologically but not smoothly isotopic. These latter surfaces are distinguished by their associated cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations. (arxiv / Quanta article)

Khovanov homology and exotic surfaces in the 4-ball
with Isaac Sundberg
J. reine angew. Math. (Crelle's), Vol. 2024, Issue 809 (2024)

We show that the cobordism maps on Khovanov homology can distinguish between exotically knotted smooth surfaces in the 4-ball (i.e., surfaces that are isotopic through ambient homeomorphisms but not ambient diffeomorphisms). This provides an elementary, combinatorial approach to the detection of exotically knotted surfaces. (arxiv / dumplings)

Project: Corks, complex curves, and exotically knotted surfaces

• Exotically knotted disks and complex curves (2020, arxiv)

• Corks, covers, and complex curves (2021, arxiv)

This pair of papers investigates the connection between surfaces in 4-manifolds that are exotically knotted (i.e., topologically but not smoothly isotopic), holomorphic disks in the 4-ball, and their connection to exotic contractible 4-manifolds known as corks. The main results construct exotic surfaces (of all genera) in the 4-ball and use this local perspective to study surfaces in larger 4-manifolds. Other applications include the first examples of exotically knotted complex curves (including proper curves in ℂ²) and symplectic 2-spheres, as well as progress on the geography/botany problem for higher-dimensional knot groups (such as exotic surfaces whose knot groups are not expected to be "good" in the sense of surgery theory).

The trace embedding lemma and spinelessness
with Lisa Piccirillo
Journal of Differential Geometry  (to appear)

This paper studies the interplay between piecewise-linear surfaces, handle structures, and smooth structures on 4-manifolds. Our main construction yields pairs of exotic 4-manifolds X and X' such that X is the knot trace of a slice knot yet X is not diffeomorphic to any knot trace. We give several related results and applications, including a sweeping family of counterexamples to Problem 4.25 in Kirby's list, providing an alternative to Levine and Lidman's recent solution. We also show that all smooth 4-manifolds contain topological locally flat surfaces that cannot be approximated by piecewise-linear surfaces, and give new results concerning the handle structures of simply-connected 4-manifolds. (arxiv)

Exotic Mazur manifolds and knot trace invariants
with Thomas E. Mark and Lisa Piccirillo
Advances in Mathematics, Vol 391 (2021)

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our obstruction comes from the knot Floer homology concordance invariant ν, which we prove is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also resolve an open question by showing that the concordance invariants τ and ϵ are not invariants of such 4-manifolds. As another application, we produce integer homology 3-spheres admitting two distinct surgeries to S¹×S², resolving a question from Problem 1.16 in Kirby's list. (arxiv, slides)

Quasipositive links and Stein surfaces
Geometry & Topology, Vol 25 (2021), 1441-1477

We study the generalization of quasipositive links from the 3-sphere to arbitrary closed, orientable 3-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provides the first half of a topological characterization of links in 3-manifolds which bound complex curves in a Stein filling. Our arguments replace pseudoholomorphic curve techniques with a study of characteristic and open book foliations on surfaces in 3- and 4-manifolds. (arxiv)

Link homologies and knotted surfaces
in New Structures in Low-Dimensional Topology, Bolyai Soc. Math. Stud., (expected 2026)
(Draft)

Link homology theories (such as knot Floer homology and Khovanov homology) have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions. These lecture notes, based on a lecture series at the Georgia Topology Conference Summer School in 2024, discuss what these toolkits say about surfaces in 4-space themselves (via the homomorphisms assigned to link cobordisms). We begin with a brief overview of these theories that focuses on their shared formal properties, then survey some of their applications to knotted surfaces. Afterwards, we give an introduction to Khovanov homology (with an eye towards its cobordism maps), discuss hands-on computational techniques for Khovanov and Bar-Natan homology, and outline the role of the Bar-Natan category in this story. 

A knot-theoretic tour of dimension four
with Márton Beke
in Singularities and Low Dimensional Topology, Bolyai Soc. Math. Stud., Vol. 30 (2024)
(arxiv)

These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the Rényi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.