April 11, 2024: Mike Wong "Knot Floer homology, barcodes, and primality"

Abstract: In this talk, we will discuss work in progress on applying the methods of persistent homology to knot Floer homology. As an application, we will look at using knot Floer homology as a primality test, including a three-line proof of Krcatovich’s result that L-space knots are prime. This is joint work with Samantha Allen, Chuck Livingston, and Misha Temkin.

February 29, 2024: Mike Miller Eismeier "Instantons, indefinite 4-manifolds, and Dehn surgery"

Abstract: In joint work with Ali Daemi, we investigate an invariant q_3(Y) in Z associated to a homology 3-sphere, and a mod-2 instanton homology analogue of the d-invariant in Heegaard Floer theory. Whereas the d-invariant is nondecreasing under negative-definite cobordisms, q_3 satisfies an inequality valid for all cobordisms W: Y -> Y' with H_1(W; Z) free of 2-torsion, even indefinite ones: we have -b^+(W) <= q_3(Y') - q_3(Y) <= b^-(W). As an application, we show the existence of hyperbolic homology spheres which are not Dehn surgery on a link of fewer than n components for any n.

February 22, 2024: Ina Petkova "Spectral GRID invariants and Lagrangian cobordisms"

Abstract: Knot Floer homology is a powerful invariant of knots and links, developed by Ozsvath and Szabo in the early 2000s. Among other properties, it detects the genus, detects fiberedness, and gives a lower bound to the 4-ball genus. The original definition involves counting homomorphic curves in a high-dimensional manifold, and as a result the invariant can be hard to compute. In 2007, Manolecu, Ozsvath, and Sarkar came up with a purely combinatorial description of knot Floer homology for knots in the 3-sphere, called grid homology. Soon after, Ozsvath, Szabo, and Thurston defined invariants of Legendrian knots using grid homology. We show that the filtered version of these GRID invariants, and consequently their associated invariants in a certain spectral sequence for grid homology, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure, strengthening a result of Baldwin, Lidman, and Wong. This is joint work with Jubeir, Schwartz, Winkeler, and Wong.

February 15, 2024: Andrew Hanlon "Resolutions from symplectic topology"

Abstract: Algebraic objects are often studied by building every element from a generating set and examining how these generators interact. We will see how such "resolutions" of objects can be understood geometrically in certain Fukaya categories and their relation to classical geometric constructions building a shape out of simpler pieces. Moreover, these geometric resolutions can be applied to obtain new algebraic results and insight via mirror symmetry. This talk is based on joint work with Hicks and Lazarev.

February 8, 2024: Oleg Lazarev "Weinstein domains: a symplectic geometer's handlebodies"

Abstract: Weinstein domains are symplectic analogs of smooth handlebodies and come equipped with decompositions with elementary symplectic pieces. As a result, they are easy to construct, have computable invariants, and include classical examples like cotangent bundles. After giving some background, I will survey several questions (and recent answers) about Weinstein domains, many of which are motivated by analogous questions in smooth topology and have categorical interpretations. For example, the minimal number of Weinstein handles in a Weinstein domain is related to the Grothendieck group of its Fukaya category while the minimal number of "elementary" Weinstein sectors needed to cover a Weinstein domain is related to the Rouquier dimension of its Fukaya category; in the case of the cotangent bundle of M, this Rouquier dimension is bounded by the LS-category of M.

January 11, 2024: Tanushree Shah "Tight contact structures on Seifert fibered 3-manifolds."

Abstract: I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3 manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory which help us calculate the lower bound and upper bound of a number of tight contact structures. We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.

November 16, 2023: Sergey Melikhov " On the analogues of Cochran's derived invariants for links of linking number 1"

Abstract: Let L=(K_1,K_2) be a two-component smooth link in S^3. In the case lk(L)=0, S. Kojima and M. Yamasaki (1979) introduced a useful invariant \eta_L, namely, the generating function of the linking numbers between a lift of a zero pushoff of K_2 and the various lifts of K_2 itself in the infinite cyclic cover of S^3-K_1. (Here the "linking number" has the usual meaning when K_1 is the unknot, and in general it is defined using that every 1-cycle in \widetilde{S^3-K_1} becomes a boundary upon being multiplied by some Laurent polynomial.) T. Cochran (1985) discovered that \eta_L is equivalent by a change of variable to a power series whose coefficients \beta^i(L) have a simple and beautiful geometric description in terms of iterated intersections of Seifert surfaces. It further turns out that these integers \beta^i(L) are in fact some of the coefficients of the power series \nabla_L(u,v,w)/\nabla_{K_1}(u), where \nabla_{K_1} is the Conway polynomial and \nabla_L is the so-called two-variable Conway polynomial. This observation of mine (2003) was originally a consequence of a theorem of J. T. Jin (1988), who expressed \eta_L in terms of the Alexander polynomials of L and K_1 by a clever argument involving many surgeries of S^3. But recently I found a simpler proof of this fact, in terms of Seifert matrices.
Cochran's derived invariants \beta^i(L), being coefficients of the power series \nabla_L(u,v,w)/\nabla_{K_1}(u), are defined for two-component smooth links of any linking number. But in the case lk(L)\ne 0 their geometry was a complete mystery until recently. I will discuss a theorem which sheds some light on their geometry in the case lk(L)=1.
One reason to be concerned about the geometry of \beta^i(L) in the case lk(L)=1 is that these invariants have been used in the recent progress on Rolfsen's problem: "Is every knot isotopic to the unknot? In particular, is the Bing sling isotopic (=homotopic through embeddings) to the unknot (or, equivalently, to a smooth knot)?". The theorem says that there exists no isotopy from the Bing sling to the unknot that extends to an isotopy of a two-component link with lk=1.

October 26, 2023: Ivan Dynnikov "Rectangular diagrams of taut foliations in knot complements"

Abstract: Taut foliations are an important instrument in low-dimensional topology. In particular, due to the works of W.Thurston and D.Gabai, they can be used to certify knot genus. Jointly with Mikhail Chernavskikh we have worked out a universal way to represent taut foliations in knot complements by using the formalism of rectangular diagrams, and shown that any finite depth taut foliation can be represented in this way. This will be explained in the talk. The work is supported by the Russian Science Foundation under grant 22-11-00299

October 12, 2023: Matt Hogancamp " Khovanov homology and handleslides "

Abstract: If K is a framed knot and X is a link in the solid torus, then the satellite operation produces a link K(X), obtained by embedding X (the "pattern") into a tubular neighborhood of K (the "companion"). In the context of Khovanov homology, the patterns can be thought of as objects of a category called the annular Bar-Natan category (ABN), and the satellite operation defines a functor from ABN to bigraded vector spaces, sending a pattern X to Kh(K(X)). In this talk I will discuss joint work with Dave Rose and Paul Wedrich, in which we construct an object Ω (which we call a "Kirby color") in ABN such that Kh(K(Ω)) is invariant under handleslides. As I will explain, the object Ω encodes the Manolescu-Neithalath 2-handle formula for the sl(2) skein lasagna modules (which was inspirational for our work). Time permitting, I will discuss an intriguing description of the Kirby object in terms of some special braids (positive braid lifts of n-cycles) that we speculate may be more amenable for computation.

September 28, 2023: Matthew Harper "Seifert-Torres Formulas for the Alexander Polynomial of Links from Quantum sl2"

Abstract: In this talk, I will recall how the Alexander polynomial, a classical knot invariant, can be constructed as a quantum invariant from quantum sl2 at a fourth root of unity. I will then discuss the development of a diagrammatic calculus based on further investigation of quantum sl2 representations. Applying this calculus in the context of the Alexander polynomial allows us to compute the invariant for certain families of links using quantum algebraic methods, rather than using methods of classical topology.

September 14, 2023: Evgeniy Scepin "Application of topology to optical recognition of handwritten characters"

Abstract: Every handwritten symbol consists of several lines. These lines may intersect. Therefore, each symbol corresponds to a graph whose vertices correspond to the intersection points of the lines and their ends, and the edges correspond to the line segments enclosed between the vertices. A scanned image of a symbol representing a Boolean matrix is fed to the input of the recognition program. On the image matrix, the lines become "thick", interference also occurs there. The report will tell you how to reduce the image so as to get a correct graph of the symbol, whose edges on the matrix are represented by thin (pixel-thick) lines that preserve the shape of the original ones.