Abstract: We will discuss the nonorientable surfaces that torus knots bound. We use a surface construction introduced by Josh Batson together with tools from knot Floer homology to compute the nonorientable four-genus of infinite families of torus knots. Comparing this surface construction with the surfaces realizing torus knots' non-orientable three-genus, we show that the difference between nonorientable three- and four-genus can be arbitrarily large. This contrasts with the analogous situation in the orientable world. Kronheimer and Mrowka proved in 1993 that both the orientable three-genus and the orientable four-genus for T(p,q) are equal to (p-1)(q-1)/2. This is joint work with Stanislav Jabuka.
May 16, 2019: Jody Trout "On the isomorphism between geometric K-homology and the topological K-homology of the Bott spectrum "
Abstract: Let X be a topological space. Using ideas of Paul Baum, we will show that there is a map, constructed at the level of cycles, which goes from the topological K-homology of X (using the Bott spectrum) to the Baum-Douglas geometric K-homology of X (using spin-c-manifolds and vector bundles), which provides the inverse map to the functorial isomorphism (due to Martin Jakob). This is achieved by simplifying the stable homotopy direct limit construction of topological K-homology into generators and relations that dispenses with the Bott spectrum.
Abstract: We show that the homology cobordism group of integer homology three-spheres contains an infinite rank summand. The proof uses an algebraic modification of the involutive Heegaard Floer package of Hendricks–Manolescu and Hendricks–Manolescu–Zemke. This is inspired by Hom's techniques in the setting of knot concordance. This is joint work with Irving Dai, Jen Hom and Matt Stoffregen.
Abstract: In 2012, Lipshitz-Sarkar introduced a stable homotopy type knot invariant whose homology recovers Khovanov homology, and showed that it is a stronger invariant than Khovanov homology alone. We show how to extract novel invariants for annular links from this stable homotopy information, and present applications to the braid conjugacy problem and to non-classical invariants of transverse links.
Abstract: We investigate the problem of determining when a cyclic branched cover of a link can be an L-space and show, for instance, that if the n-fold branched cyclic cover of an L-space knot is an L-space, then n is at most 5. Along the way we show how this problem leads to the calculation of the smooth and topological 4-ball genera of quasi-alternating quasipositive links. This is joint work with Michel Boileau and Cameron Gordon.
Abstract: I will review Lorentz manifolds as a model of space-time, and how the important notion of causal structure arises in this context. Then, observing the fundamental nature of null geodesics to this structure, I will describe the space of null geodesics and the natural topological and geometric structures it carries. Finally, I will consider how aspects of the causal structure of the original space-time is encoded in this space of null geodesics, culminating in the notion of Legendrian linking of those submanifolds in the space of null geodesics representing points of the space-time.
March 28, 2018: Artem Kotelskiy "Khovanov homology and the Fukaya category of the 4-punctured sphere"
Abstract: Consider a 2-sphere S intersecting a knot in 4 points. This defines a decomposition of a knot into two 4-ended tangles K=T+Q. We will show that Khovanov homology Kh(K), and its deformation due to Bar-Natan, are isomorphic to Lagrangian Floer homology of specifically constructed immersed curves on the initial 4-punctured sphere S. This result is analogous to immersed curves description of bordered Heegaard Floer homology and knot Floer homology. The key step will be defining a certain cobordism theoretic algebra B, which is a deformation of Khovanov's arc algebra. We will show that B embeds in a nice way into the wrapped Fukaya category of the 4-punctured sphere Fuk(S). This, together with the cobordism theoretic tangle invariants due to Bar-Natan, and the classification of objects inside Fuk(S), will be the main parts of the construction. This is joint work with Liam Watson and Claudius Zibrowius.
March 7, 2018: Akram Alishahi "Braid invariant related to knot Floer homology and Khovanov homology"
Abstract: Khovanov homology and knot Floer homology are two knot invariants that were defined around the same time, and despite their different constructions, share many formal similarities. After reviewing the construction of Khovanov homology and some of these similarities, I will sketch the definition of an algebraic braid invariant which is closely related to both Khovanov homology and the refinement of knot Floer homology into tangle invariants. This is a joint work with Nathan Dowlin.
Abstract: p-DG theory is a useful framework for categorifying quantum groups and their representations at prime roots of unity. We review this structure and then apply this machinery in the context of a specific categorical braid group action.
Abstract: For Legendrian and transverse links in the 3-sphere, Ozsvath, Szabo, and Thurston defined combinatorial invariants that reside in grid homology. Known as the GRID invariants, they are effective in distinguishing some transverse knots that have the same classical invariants. In this talk, we describe some recent developments: First, we show that the GRID invariants obstruct decomposable Lagrangian cobordisms; second, we outline a computable generalization via cyclic branched covers. The first result is joint with John Baldwin and Tye Lidman, and the second with Shea Vela-Vick.
Abstract: A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of n or fewer crossings approaches 1 as n approaches infinity. We show that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.
Abstract: A singular virtual 2-string $\alpha$ is a wedge of two circles on a closed oriented surface. Up to equivalence by virtual homotopy, $\alpha$ can be realized on a canonical surface $\Sigma_\alpha$. We use the homological intersection pairing on $\Sigma_\alpha$ to associate an algebraic object to $\alpha$ called a singular based matrix. In this talk, we show that these objects can be used to distinguish virtual homotopy classes of 2-strings and to compute the virtual Andersen--Mattes--Reshetikhin bracket of families of 2-strings.
Abstract: A knot is a circle in 3-space. A main problem in knot theory is distinguishing knots (two knots are equivalent if we can continuously deform one into the other). One way to approach this is by studying algebraic "knot invariants" — algebraic objects associated to knots, which do not change as the knot is deformed. In 1928, J. Alexander described a knot invariant, now called the Alexander polynomial. In the early 2000s, Ozsvath and Szabo constructed a powerful refinement of the Alexander polynomial, called knot Floer homology (HFK). Among other properties, it detects the genus, detects fiberedness, and gives a lower bound on the 4-ball genus. The original definition involves counting holomorphic curves in a high-dimensional manifold, and as a result can be hard to compute.
Tangle Floer homology is a new algebraic technique for studying HFK, by cutting a knot into pieces called tangles, and studying the individual pieces and their gluing. One associates a differential graded algebra (DGA) to a sequence of points, and a dg bimodule over the respective DGAs to a tangle with two sets of "ends". Given a decomposition of a knot into tangles, the derived tensor product of the bimodules associated to the pieces recovers the knot Floer homology of the glued knot. After providing a bit of general background, we'll try to sketch out a purely combinatorial definition of this invariant.
Abstract: The nonorientable 4–genus of a knot K is the minimal first Betti number of a nonorientable surface in B^4 whose boundary is K. Finding the nonorientable 4–genus of a knot can be quite intractable; existing methods exploit the relationship between nonorientable genus and normal Euler number of the nonorientable surface. In this talk, I will give an overview of the interplay between the nonorientable genus and normal Euler number of nonorientable surfaces in B^4. I will define both of these invariants and discuss their computation. In particular, when fixing a knot K, we can ask what pairs of nonorientable genus and normal Euler number are realizable for a surface whose boundary is K. We will see that both classical invariants and Heegaard–Floer invariants can be used towards answering this question.
Abstract: Most of the 50-year history of the study of the set of smooth knot concordance classes, C, has focused on its structure as an abelian group. Tim Cochran and I took a different approach, namely we studied C as a metric space (with the slice genus metric) admitting many natural geometric operators. The goal was to give evidence that the knot concordance is a fractal space. However, both of these metrics are integer valued metrics and so induce the discrete topology. Subsequently, with Mark Powell, we defined a family of real valued metrics, called the q-grope metrics, that take values in the real numbers and showed that there are sequences of knots whose q-norms get arbitrarily small for q>1. However, for q>1, this metric vanishes on topologically slice knots (it is really a pseudo metric). In this talk, we define a new metric (called the tower metric) based on a new objects which we called positive and negative towers, using a combination of generalized handles and gropes. This is an interesting metric since it is related to the bipolar filtration, a filtration generalizing work of Gompf and Cochran. Using recent work of Cha and Kim on the non-triviality of the bipolar filtration of the group of topologically slice knots, we show that there are sequences of topologically slice knots whose q-norms get arbitrarily small but are never 0. This work is joint with Tim Cochran, Mark Powell, and Aru Ray.
Abstract: Let K be a link braided about an open book (B,p) supporting a contact manifold (Y,xi). K and B are naturally transverse links. We prove that the hat version of the transverse link invariant defined by Baldwin, Vela-Vick and Vertesi is non-zero for the union of K with B. As an application, we prove that the transverse invariant of any braid having fractional Dehn twist coefficient greater than one is non-zero. This generalizes a theorem of Plamenevskaya about classical braid closures.