Braids Reunion Workshop
Institute for Computational and Experimental Research in Mathematics (ICERM)
July 15, 2024  July 19, 2024
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Monday, July 15, 2024

9:20  9:30 am EDTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

9:30  10:15 am EDTShortest word problem in braid theory11th Floor Lecture Hall
 Speaker
 Keiko Kawamuro, University of Iowa
 Session Chair
 Nancy Scherich, Elon University
Abstract
Given a braid element in B_n, searching for a shortest braid word representative (using the bandgenerators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele CapovillaSearle.

10:30  11:00 am EDTCoffee Break11th Floor Collaborative Space

11:00  11:45 am EDTSymmetries of link homology11th Floor Lecture Hall
 Speaker
 Joshua Sussan, CUNY
 Session Chair
 Nancy Scherich, Elon University
Abstract
We construct an action of sl(2) on equivariant KhovanovRozansky link homology.
This is joint with You Qi, LouisHadrien Robert, and Emmanuel Wagner. 
12:00  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTBiordering link complements via braids11th Floor Lecture Hall
 Speaker
 Hannah Turner, Stockton University
 Session Chair
 Caitlin Leverson, Bard College
Abstract
Any link (or knot) group – the fundamental group of a link complement – is leftorderable. However, not many link groups are biorderable – that is, admit an order invariant under both left and right multiplication. It is not well understood which link groups are biorderable, nor is there is a conjectured topological characterization of links with biorderable link groups. I will discuss joint work in progress with Jonathan Johnson and Nancy Scherich to study this problem for braided links – braid closures together with their braid axis. Inspired by KinRolfsen, we focus on braided link groups because algebraic properties of the braid group can be employed in this setting. In particular, I will discuss our implementation of an algorithm which, given a braided link group which is not biorderable, will return a definitive "no" and a proof in finite time. Using our program, we give a new infinite family of nonbiorderable braided links.

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTUniform twisted homological stability of braid groups and moments of quadratic Lfunctions11th Floor Lecture Hall
 Speaker
 Peter Patzt, University of Oklahoma
 Session Chair
 Caitlin Leverson, Bard College
Abstract
A conjecture of ConreyFarmerKeatingRubinsteinSnaith aims to describe the asymptotics of moments of quadratic Lfunctions. In joint work with Miller, Petersen, and RandalWilliams and in combination with a paper by Bergström–Diaconu–Petersen–Westerland, we proved a version of this conjecture for function fields. Using the GrothendieckLefschetz trace formula, Bergström–Diaconu–Petersen–Westerland showed a connection between the conjecture and the twisted homology of the braid groups. In our paper, we showed what was needed to make this connection. Homological stability says that the kdimensional homology groups are all isomorphic for a large enough number of strands of the braid groups. This is even known for twisted coefficients pulled back from polynomial representations of the symplectic groups. We proved that the starting point of stability is independent of which irreducible polynomial representation of the symplectic groups one uses. In the talk, I will explain the connections between number theory, the braid groups, the symplectic groups, and homological stability.

4:30  6:00 pm EDTReception11th Floor Collaborative Space
Tuesday, July 16, 2024

9:30  10:15 am EDTSkein Lasagna Modules and Categorified Projectors11th Floor Lecture Hall
 Speaker
 Melissa Zhang, University of California, Davis
 Session Chair
 Miriam Kuzbary, Amherst College
Abstract
In 2018, Morrison, Walker, and Wedrich’s skein lasagna modules are 4manifold invariants defined using KhovanovRozansky homology similarly to how skein modules for 3manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2handlebodies by defining an isomorphic object called cabled KhovanovRozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4manifold.
In joint work with Ian Sullivan, we lift the ManolescuNeithalath construction to the level of BarNatan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in BarNatan's category. As an application, we give a proof that the skein lasagna module of S2xS2 is trivial, confirming a conjecture of Manolescu. Our local techniques also allow for computations of the skein lasagna invariant for other 4manifolds whose Kirby diagram contains a 0framed unknot component. Our methods also allow us to relate the RozanskyWillis invariant of links in S2xS1 to skein lasagna modules. 
10:30  11:00 am EDTCoffee Break11th Floor Collaborative Space

11:00  11:45 am EDTCharacterizing and noncharacterizing knots by 3manifolds11th Floor Lecture Hall
 Speaker
 Marc Kegel, HumboldtUniversität zu Berlin
 Session Chair
 Miriam Kuzbary, Amherst College
Abstract
From a knot K, we can build 3manifolds by performing Dehn surgery on that knot. We will discuss some new results explaining in which sense the diffeomorphism types of these 3manifolds characterize the isotopy class of the knot K. This talk is based on joint work with AbeWeiss, Baker, BakerMcCoy, CasalsEtnyre, and Piccirillo.

12:00  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTDetecting corks11th Floor Lecture Hall
 Speaker
 Abhishek Mallick, Rutgers University  New Brusnwick
 Session Chair
 Nicolas Petit, Loyola University Chicago
Abstract
Corks are fundamental to the study of exotic smooth structures on 4manifolds. In this talk, I will describe how to detect corks and their usefulness. This is joint work with many collaborators over several projects.

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTA topological model for the HOMFLYPT polynomial11th Floor Lecture Hall
 Speaker
 Christine Ruey Shan Lee, Texas State University
 Session Chair
 Nicolas Petit, Loyola University Chicago
Abstract
A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLYPT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and SL_n polynomials, and the construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLYPT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.
Wednesday, July 17, 2024

9:30  10:15 am EDTtba11th Floor Lecture Hall
 Speaker
 İnanç Baykur, University of Massachusetts Amherst
 Session Chair
 Orsola CapovillaSearle, University of California, Davis

10:30  11:00 am EDTCoffee Break11th Floor Collaborative Space

11:00  11:45 am EDTTBA11th Floor Lecture Hall
 Speaker
 Siddhi Krishna, Columbia University
 Session Chair
 Orsola CapovillaSearle, University of California, Davis

12:00  12:05 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:00  2:00 pm EDTProblem Session in small groups over catered lunchLunch/Free Time  11th Floor Lecture Hall

2:00  2:45 pm EDTNonorientable broken Lefschetz fibrations11th Floor Lecture Hall
 Speaker
 Porter Morgan, University of Massachusetts Amherst
 Session Chair
 Marc Kegel, HumboldtUniversität zu Berlin
Abstract
Broken Lefschetz Fibrations (BLFs) are surjections from a 4manifold to a sphere with only Lefschetz and indefinite fold singularities. Unlike Lefschetz fibrations, any generic map M^4\to S^2 is homotopic to a BLF. Among BLFs, a particularly nice subset is Simplified Broken Lefschetz Fibrations, which give an explicit handlebody decomposition of their source manifold. In this talk, we’ll review the basic topology of simplified BLFs for closed, smooth, nonorientable 4manifolds. Then we’ll use simplified BLFs to construct an explicit embedding of nonorientable 4manifolds into certain 6manifolds.

3:00  3:30 pm EDTYellow Pig Day Coffee BreakCoffee Break  11th Floor Collaborative Space

3:00  4:30 pm EDTDesignated collaboration/ work in groups timeGroup Work  11th Floor Collaborative Space
Thursday, July 18, 2024

9:30  10:15 am EDTCorrection terms of branched double covers and symmetries of immersed curves11th Floor Lecture Hall
 Speaker
 Biji Wong, Duke University
 Session Chair
 Siddhi Krishna, Columbia University
Abstract
In this talk, we'll discuss recent work to use the immersed curves description of bordered Floer theory to study the dinvariants of branched double covers Sigma_2(L) of links L in the 3sphere. We'll show that when L is a 2component plumbing link and Sigma_2(L) is an Lspace, then the spin dinvariants of Sigma_2(L) are determined by the signatures of L. This project is joint with J. Hanselman and M. Marengon.

10:30  11:00 am EDTCoffee Break11th Floor Collaborative Space

11:00  11:45 am EDTContact surgery numbers11th Floor Lecture Hall
 Speaker
 Rima Chatterjee, University of Cologne
 Session Chair
 Siddhi Krishna, Columbia University
Abstract
A fundamental result in 3dimensional contact topology due to DingGeiges tells us that any contact 3manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a particular contact manifold? Contact surgery number is a measure of this complexity. In this talk, I will define this notion of complexity and discuss some examples. This is joint work with Marc Kegel.

12:00  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTExact Lagragian fillings of Legendrian links and Newton polytopes.11th Floor Lecture Hall
 Speaker
 Orsola CapovillaSearle, University of California, Davis
 Session Chair
 Biji Wong, Duke University
Abstract
An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about connections between fillings and Newton polytopes.

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTHighdimensional antisurgery for Weinstein manifolds11th Floor Lecture Hall
 Speaker
 Angela Wu, Louisiana State University
 Session Chair
 Biji Wong, Duke University
Abstract
A Legendrian knot in the boundary of a Weinstein domain of dimension at least 6 which bounds a Lagrangian disk can be considered the boundary of the cocore of a handle. A Weinstein antisurgery amounts to carving out this handle from the Weinstein domain. In this talk, I’ll explain an algorithm which constructs explicit handle decompositions of many of these highdimensional Weinstein antisurgery manifolds using a new highdimensional Legendrian isotopy. I’ll give a specific application of this algorithm to Lazarev and Sylvan’s class of Weinstein manifolds which they called Pflexible, formed from handle attachment along Ploose Legendrians. This talk is based on work in progress with Ipsita Datta, Oleg Lazarev, and Chindu Mohanakumar.
Friday, July 19, 2024

9:30  10:15 am EDTWork in progress — a gaugetheoretic interpretation of the McKay correspondence11th Floor Lecture Hall
 Speaker
 Jiajun Yan, University of Virginia
 Session Chair
 Jonathan Johnson, Oklahoma State University
Abstract
Let Gamma be a finite subgroup of SU(2). The McKay correspondence states that the McKay quiver of Gamma is isomorphic to the graph of minimal resolution of C^2/Gamma. There are various proofs of the McKay correspondence coming from algebraic geometry and symplectic geometry which we will survey in the talk. Then, we will present an idea / work in progress of a new gaugetheoretic interpretation of the McKay correspondence. To do so, we will review a gaugetheoretic construction of the 4dimensional hyperkähler ALE spaces for each of which the underlying topological space is the minimal resolution of C^2/Gamma. The main approach is to make use of an S^1invariant MorseBott function arising from the gaugetheoretic construction by identifying its critical points with certain flat connections that induce representations of Gamma via holonomy representation. The idea is partially inspired by the ChernSimons theory, and some tools from contact geometry also come into play.

10:30  11:00 am EDTCoffee Break11th Floor Collaborative Space

11:00  11:45 am EDTNew quantum invariants from braiding Verma modules11th Floor Lecture Hall
 Speaker
 Sergei Gukov, Caltech
 Session Chair
 Jonathan Johnson, Oklahoma State University
Abstract
In this talk, I will describe recent construction of new link and 3manifold invariants associated with Verma modules of $U_q (sl_N)$ at generic $q$. The resulting invariants can be combined into a Spin$^c$decorated TQFT and have a nice property that, for links in general 3manifolds, they have integer coefficients. In particular, they are expected to admit a categorification and, if time permits, I will outline various ingredients that may go into a construction of 3manifold homology categorifying $U_q (sl_N)$ invariants at generic $q$.

12:00  2:00 pm EDTLunch/Free Time
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