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Organizing Committee
Amherst College
Harvard University
University of Glasgow
University of Virginia
Abstract
A topologist might hope that results in knot theory can always be extended to links. Decades of work in link theory show such extensions are not always straightforward and are, in fact, sometimes impossible. There are also properties and strategies unique to link theory. For all these reasons, link theory is a rich source of relationships to driving questions in the study of topology in dimensions 3 and 4. Among the most compelling are connections to 4-dimensional surgery, exotica in dimension 4, and the study of surfaces in 4-manifolds. Additionally, a vast number of open questions and unexplored topics remain within the confines of the theory in areas such as link concordance, link homotopy, and homology theories, including Heegaard Floer homology and Khovanov homology.
This workshop will bring together low-dimensional topologists of all backgrounds to further the general knowledge of link theory within the low-dimensional topology community, including techniques and tools used to study links and important open questions connected to link theory. In particular, the workshop will highlight how computation interacts with current research in link theory. The workshop will also provide a space for new collaborations to develop and promote a supportive and inclusive environment within the community of researchers.
This workshop will bring together low-dimensional topologists of all backgrounds to further the general knowledge of link theory within the low-dimensional topology community, including techniques and tools used to study links and important open questions connected to link theory. In particular, the workshop will highlight how computation interacts with current research in link theory. The workshop will also provide a space for new collaborations to develop and promote a supportive and inclusive environment within the community of researchers.
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Workshop Schedule
Monday, May 12, 2025
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Roots of Alexander polynomials of random positive 3-braids
11th Floor Lecture Hall
Speaker
Nathan Dunfield, University of Illinois, Urbana-Champaign
Session Chair
Brendan Owens, University of Glasgow
Abstract
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. This is joint work with Giulio Tiozzo.
10:00 - 10:30 AM EDT
Coffee Break
10th Floor Collaborative Space
10:30 - 11:15 AM EDT
Colored Heegaard Floer homology: Structures and Examples
11th Floor Lecture Hall
Speaker
Beibei Liu, The Ohio State university
Session Chair
Brendan Owens, University of Glasgow
Abstract
We define new knot invariants, called colored Heegaard Floer homology for any knot K in the three sphere. This is the colimit of the cobordism maps from the (n, mn)-cable of K to (n, (m+1)n)-cable by letting m goes to infinity which are invariant under full twists. We will describe the module structures of the colored Heegaard Floer homology and show that the colimit is bounded in each gradings.
11:30 AM - 12:15 PM EDT
A higher order torsion linking form and 3-manifolds in 4-space
11th Floor Lecture Hall
Speaker
Slava Krushkal, University of Virginia
Session Chair
Brendan Owens, University of Glasgow
Abstract
Given a rational homology 3-sphere, we introduce a triple linking form on its first homology, defined when the classical torsion linking pairing of three homology classes vanishes pairwise. We use these methods to formulate an embedding obstruction for rational homology spheres in S^4, extending a 1938 theorem of Hantzsche. (Based on joint work with Michael Freedman)
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Higher order Kirk invariants of knotted spheres in 4-space
11th Floor Lecture Hall
Speaker
Jean-Baptiste Meilhan, Université Grenoble Alpes
Session Chair
Hannah Turner, Stockton University
Abstract
Paul Kirk defined in the late eighties a link-homotopy invariant of 2-component link maps, which are 'self-immersed spheres' in 4-space. Recently, Schneiderman and Teichner showed that this 'Kirk invariant' is in fact a complete link-homotopy invariant. In this talk, I will explain how to extend Kirk's construction to link maps of any number of components, in an explicit and computable way. This is a joint work with B. Audoux and A. Yasuhara.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 5:00 PM EDT
Open Problem Session
Problem Session - 11th Floor Lecture Hall
Session Chair
Hannah Turner, Stockton University
Moderator
Shelly Harvey, Rice University
Moderator
Constance Leidy, Wesleyan University
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, May 13, 2025
9:00 - 9:45 AM EDT
SL(4) Foam Evaluation and 6-Valent Seams
11th Floor Lecture Hall
Speaker
Haihan Wu, Johns Hopkins University
Session Chair
Sumeyra Sakalli, University of South Florida
Abstract
A crucial step to categorification of link (or web) polynomials via the `tautological construction' is the evaluation of closed foams. Robert-Wagner's `SL(n)' foam evaluations provide a purely combinatorial method for computing SL(n) link homology. In practice, the foams they use are actually GL(n) foams, which have facets of weights 1 through 4.
Inspired by the recent development on rotational-invariant SL(4) web basis by Gaetz-Pechenik- Pfannerer-Striker-Swanson, we modify the Robert-Wagner foam evaluation formula to accommodate the use of SL(4) foams, which now feature only facets of weights 1 and 2, at the cost of introducing new types of defect lines and vertices. Additionally, we introduce a categorification of the `hexagon web relation' for both GL(4) and SL(4) foams.
In this talk, I will review background material on webs and foams and talk about the modified SL(4) foam construction. If time permits, I will also mention its connections to SL(4) web algebra and SL(4) affine building. This talk is based on upcoming joint work with Jernej Grlj, Mikhail Khovanov, and Melissa Zhang.
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