Braids Reunion Workshop

Institute for Computational and Experimental Research in Mathematics (ICERM)

https://icerm.brown.edu/topical_workshops/tw-24-braids
July 15, 2024 - July 19, 2024
Monday, July 15, 2024
  • 9:20 - 9:30 am EDT
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 9:30 - 10:15 am EDT
    Shortest word problem in braid theory
    11th 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 band-generators) 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 Capovilla-Searle.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    tba
    11th Floor Lecture Hall
    • Speaker
    • Joshua Sussan, CUNY
    • Session Chair
    • Nancy Scherich, Elon University
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Bi-ordering link complements via braids
    11th 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 left-orderable. However, not many link groups are bi-orderable – that is, admit an order invariant under both left and right multiplication. It is not well understood which link groups are bi-orderable, nor is there is a conjectured topological characterization of links with bi-orderable 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 Kin-Rolfsen, 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 bi-orderable, will return a definitive "no" and a proof in finite time. Using our program, we give a new infinite family of non-bi-orderable braided links.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Uniform twisted homological stability of braid groups and moments of quadratic L-functions
    11th Floor Lecture Hall
    • Speaker
    • Peter Patzt, University of Oklahoma
    • Session Chair
    • Caitlin Leverson, Bard College
    Abstract
    A conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith aims to describe the asymptotics of moments of quadratic L-functions. In joint work with Miller, Petersen, and Randal-Williams and in combination with a paper by Bergström–Diaconu–Petersen–Westerland, we proved a version of this conjecture for function fields. Using the Grothendieck-Lefschetz 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 k-dimensional 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 EDT
    Reception
    11th Floor Collaborative Space
Tuesday, July 16, 2024
  • 9:30 - 10:15 am EDT
    Skein Lasagna Modules and Categorified Projectors
    11th 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 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.
    In joint work with Ian Sullivan, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan'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 4-manifolds whose Kirby diagram contains a 0-framed unknot component. Our methods also allow us to relate the Rozansky-Willis invariant of links in S2xS1 to skein lasagna modules.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    Correction terms of branched double covers and symmetries of immersed curves​
    11th Floor Lecture Hall
    • Speaker
    • Biji Wong, Duke University
    • Session Chair
    • Miriam Kuzbary, Amherst College
    Abstract
    In this talk, we'll discuss recent work to use the immersed curves description of bordered Floer theory to study the d-invariants of branched double covers Sigma_2(L) of links L in the 3-sphere. We'll show that when L is a 2-component plumbing link and Sigma_2(L) is an L-space, then the spin d-invariants of Sigma_2(L) are determined by the signatures of L. This project is joint with J. Hanselman and M. Marengon.
  • 12:00 - 2:00 pm EDT
    Problem Session in small groups over catered lunch
    Lunch/Free Time - 11th Floor Lecture Hall
  • 2:00 - 2:45 pm EDT
    Cluster structures on braid varieties
    11th Floor Lecture Hall
    • Speaker
    • Jose Simental Rodriguez, Universidad Nacional Autónoma de México
    • Session Chair
    • Nicolas Petit, Loyola University Chicago
    Abstract
    In the previous 'Braids' meeting, I defined braid varieties, explained some of their properties and conjectured that they admit a cluster structure, that is, their coordinate algebra can be given the structure of a cluster algebra. I will recall the definition and examples of braid varieties, and explain how a cluster structure on them may indeed be constructed using the graphical calculus of weaves. If time permits, I will discuss how the cluster structure reflects properties of the braid and vice versa. This is based on joint works with various subsets of {Roger Casals, Marco Castronovo, Eugene Gorsky, Mikhail Gorsky, Ian Le, Linhui Shen, David Speyer}.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    A topological model for the HOMFLY-PT polynomial
    11th 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 HOMFLY-PT 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 HOMFLY-PT 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 EDT
    tba
    11th Floor Lecture Hall
    • Speaker
    • Inanc Baykur, University of Massachusetts Amherst
    • Session Chair
    • Orsola Capovilla-Searle, University of California, Davis
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    TBA
    11th Floor Lecture Hall
    • Speaker
    • Siddhi Krishna, Columbia University
    • Session Chair
    • Orsola Capovilla-Searle, University of California, Davis
  • 12:00 - 12:05 pm EDT
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:05 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Nonorientable broken Lefschetz fibrations
    11th Floor Lecture Hall
    • Speaker
    • Porter Morgan, University of Massachusetts Amherst
    • Session Chair
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    Lefschetz fibrations are a special type of map from a 4-manifold to a surface, usually either S^2 or D^2. Although they provide a lot of information about their source manifold, they’re only admitted by a limited collection of 4-manifolds. This motivates the study of broken Lefschetz fibrations; by relaxing the definition of a Lefschetz fibration, we get a family of maps that all closed, smooth 4-manifolds admit. In this talk, we’ll review the basic topology of broken Lefschetz fibrations, both in the more-established orientable case, and in the more-recent nonorientable case. We’ll examine the data that such a map provides, and see how we can obtain a broken Lefschetz fibration from a more general map onto S^2.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Designated collaboration/ work in groups time
    Group Work - 11th Floor Collaborative Space
Thursday, July 18, 2024
  • 9:30 - 10:15 am EDT
    Characterizing and non-characterizing knots by 3-manifolds
    11th Floor Lecture Hall
    • Speaker
    • Marc Kegel, Humboldt-Universität zu Berlin
    • Session Chair
    • Siddhi Krishna, Columbia University
    Abstract
    From a knot K, we can build 3-manifolds by performing Dehn surgery on that knot. We will discuss some new results explaining in which sense the diffeomorphism types of these 3-manifolds characterize the isotopy class of the knot K. This talk is based on joint work with Abe-Weiss, Baker, Baker-McCoy, Casals-Etnyre, and Piccirillo.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    Contact surgery numbers
    11th Floor Lecture Hall
    • Speaker
    • Rima Chatterjee, University of Cologne
    • Session Chair
    • Marc Kegel, Humboldt-Universität zu Berlin
    Abstract
    A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. 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 EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    TBA
    11th Floor Lecture Hall
    • Speaker
    • Orsola Capovilla-Searle, University of California, Davis
    • Session Chair
    • Biji Wong, Duke University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    TBA
    11th Floor Lecture Hall
    • Speaker
    • Angela Wu, Louisiana State University
    • Session Chair
    • Biji Wong, Duke University
Friday, July 19, 2024
  • 9:30 - 10:15 am EDT
    Work in progress — a gauge-theoretic interpretation of the McKay correspondence
    11th 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 gauge-theoretic interpretation of the McKay correspondence. To do so, we will review a gauge-theoretic construction of the 4-dimensional 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^1-invariant Morse-Bott function arising from the gauge-theoretic 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 Chern-Simons theory, and some tools from contact geometry also come into play.
  • 10:30 - 11:00 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:00 - 11:45 am EDT
    New quantum invariants from braiding Verma modules
    11th 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 3-manifold 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 3-manifolds, 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 3-manifold homology categorifying $U_q (sl_N)$ invariants at generic $q$.
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time

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