Akram Alishahi (Georgia)
Braid invariants relating Khovanov homology and knot Floer homology
I will describe an algebraic, glueable invariant for open braids and a closing off operation which results in an invariant for the plat closure of the braid. It is defined such that the plat closure invariant is the filtered chain complex which conjecturally computes HFK_2 and gives a spectral sequence whose E_2 page is Khovanov homology. On the other hand, our invariant for an open braid is in the form of a DA bimodule defined over the same algebra as the Ozsvath-Szabo's tangle invariant for computing knot Floer homology, and in fact chain homotopic to it. We will sketch this chain homotopy and compare our closing off operation with theirs. This is a joint work in progress with Nathan Dowlin.
Nathan Dowlin (Columbia)
The knot homology theory HFK_2
I will describe a family of knot invariants HFK_n which are the knot Floer analogs of Khovanov-Rozansky sl_n homology. Since sl_2 homology is isomorphic to Khovanov homology, the HFK_2 theory gives a nice framework for relating Khovanov homology and knot Floer homology. Next, I'll discuss an algebraically defined complex which conjecturally computes HFK_2, which also gives an invariant for open braids and their plat closures. The invariant for open braids is defined over the same algebra as the Ozsvath-Szabo tangle invariant for knot Floer homology. It comes equipped with a filtration, and for plat closures, the E_2 page of the spectral sequence induced by this filtration is isomorphic to Khovanov homology. Joint work with Akram Alishahi.
John Etnyre (Georgia Tech)
Contact surgery and symplectic fillability
It is fairly well known what interesting properties of a contact structure are preserved under contact (-1) surgeries (or more generally negative contact surgeries), but very little is known concerning contact (+1) surgeries. In this talk we will completely understand when contact (r) surgery is symplectically fillable for r in [0,1] and discuss possible extensions for larger r.
Matt Hedden (Michigan State)
Knot Floer homology and the relative adjunction inequality
I'll discuss the generalized tau invariants for knots in a 3-manifold Y equipped with an arbitrary non-zero Floer class. They satisfy adjunction inequalities in 4-manifolds bounded by Y whose cobordism map contains the chosen class in its image. I'll say how to prove the inequality, some words about its history, and then talk about various applications. For instance, it gives rise to a slice-Bennequin inequality for knots in a contact manifold with non-zero contact invariant. This motivates a conjectural 4-dimensional interpretation of tightness. All this is joint with Katherine Raoux. Time permitting, I'll give an independent 10 minute talk that describes through an example a mapping cone formula for the knot Floer homology of the core of surgery on a knot. This latter formula is joint with Adam Levine.
Jen Hom (Georgia Tech)
Concordance and homology cobordism homomorphisms
We define an infinite family of linearly independent integer-valued concordance and homology cobordism homomorphisms. Our concordance homomorphisms rely on knot Floer complexes over the ring F[U,V]/(UV=0), while our homology cobordism homomorphisms rely on involutive Floer complexes over F[U,Q]/(Q^2=UQ=0). We will discuss various properties and applications of these homomorphisms. This is joint work with I. Dai, M. Stoffregen, and L. Truong.
Tye Lidman (NC State)
Homology cobordisms with no 3-handles
Homology cobordisms without 3-handles arise naturally in low-dimensional topology, either through Stein cobordisms or the exteriors of ribbon concordances. We give several obstructions to the existence of such homology cobordisms, including Floer homology and SU(2)-character varieties. This is joint work with Ali Daemi, Shea Vela-Vick, and Mike Wong.
Francesco Lin (Columbia)
On Stein fillings and Pin(2)-monopole Floer homology
We discuss some applications of Pin(2)-monopole Floer homology to the topology of Stein fillings of a given rational homology sphere; in particular, we show that under simple hypotheses one can provide severe restrictions on the intersection form of its Stein fillings which are not negative definite.
Jianfeng Lin (San Diego)
On the monopole Lefschetz number of finite order diffeomorphisms
The monopole Floer homology is a powerful invariant of 3–manifolds which has had many important applications in low-dimensional topology. Because of its functoriality, the monopole Floer homology of a 3–manifold is acted upon by its mapping class group. However, the information contained in this action is not easy to extract due to the gauge theoretic nature of the theory. In this talk, we make some first steps towards understanding this action by calculating the Lefschetz numbers of any finite order diffeomorphism making the 3–manifold into branched cover over a knot in the homology 3–sphere. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology. This is a joint work with Danny Ruberman and Nikolai Saveliev.
Tom Mrowka (MIT)
Bar Natan in the wild
An interesting version of instanton Floer homology defined in characteristic 2 discovered recently with Kronheimer turns out to in certain specializations admit a spectral sequence whose E_2-term is Bar-Natan’s variant of Floer homology. Related theories give rise to a family of concordance invariants. Certain specializations gives rise to non-orientable genus bounds. I’ll try to sketch some of this story.
Lenny Ng (Duke)
An update on knot invariants from conormals
One can produce interesting invariants of knots through various Floer-theoretic counts of holomorphic curves in the cotangent bundle of R^3 bounded by the Lagrangian knot conormal or its friends. I'll survey what is currently known or conjectured about how these invariants relate to knot invariants that other people care about: the A-polynomial, colored and regular HOMFLY-PT polynomials, the Alexander polynomial, and more. The discussion is motivated by topological string theory but I'll tread lightly on this point and mainly stick to topology.
Lisa Piccirillo (Brandeis)
Exotic Mazur manifolds and property R
The simplest compact contractible 4-manifolds, other than the 4-ball, are Mazur manifolds (from a handle theoretic perspective). We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from a new repurposing of Heegaard Floer concordance invariants as smooth 4-manifold invariants. As a corollary, we produce integer homology 3-spheres admitting multiple distinct S^1 x S^2 surgeries, which gives counterexamples to a generalization of Property R, resolving a question from Problem 1.16 in Kirby's list. This is joint work in progress with Kyle Hayden and Tom Mark.
Jake Rasmussen (Cambridge)
Floer homology and representation varieties
I'll discuss some interesting parallels (and puzzling differences) between Floer homology and the SL_2(R) character varieties of knot complements. This is joint work with Nathan Dunfield.
Steven Sivek (Imperial)
Instantons and L-space surgeries
Framed instanton homology is a gauge theoretic invariant which bears a strong resemblance to the hat version of Heegaard Floer homology. In this talk we will explain how the map on framed instanton homology induced by a cobordism X decomposes into summands indexed by elements of Hom(H_2(X),Z), analogous to the spin^c decomposition of Heegaard Floer cobordism maps. We will use this decomposition to prove that "instanton L-space knots" are fibered, and then discuss applications to questions about the fundamental groups of Dehn surgeries on knots and to the A-polynomial. This is joint work with John Baldwin.
András Stipsicz (Rényi)
Concordance invariants from covering involutions
We use the branched cover construction and ideas from connected Floer homology to define concordance invariants of knot in S^3. Calculations can be performed for double branched covers, in which case the invariants are trivial for alternating and torus knots and non-trivial for some pretzel knots.
Vera Vértesi (Strasbourg)
Additivity of support norm under connected sum
The support norm of a contact structure is defined as the minimum of the negative Euler characteristics of pages of open books supporting the given contact structure. The support norm is the contact analog for the minimal Heegaard genus for smooth manifolds, and they both measure some sort of complexity of smooth or contact manifolds. The Heegaard genus of smooth manifolds, was proven to be additive under connected sum in 1968 in Haken's Lemma. In this talk I will prove the additivity of the support norm for tight contact structures. The proof uses the less known, but (by now) classical toolset of open book foliations, first invented by Bennequin in 1982 in his PhD thesis. The method used in the proof suggests a new infinite series of candidates for contact manifolds with possibly arbitrarily high support genus, and I will describe them if time permits.
Liam Watson (British Columbia)
Cabling in terms of immersed curves
I will describe the effect of cabling in knot Floer homology in terms of how this operation changes the immersed-curves interpretation of bordered Floer homology. This gives a quick-route to certain now well-known formulas due to Hedden, Hom, and others for the behaviour of Floer-theoretic concordance invariants, and results from a combination of joint work with Hanselman and Rasmussen, together with earlier work with Hanselman.
Ian Zemke (Princeton)
Doubling tricks and knot Floer homology
Given a concordance C from K_0 to K_1, one can form a concordance from K_0 to K_0 by doubling C. A Morse function on C gives a simple description of the doubled concordance in terms of adding tubes and tubing on spheres, which in turn gives algebraic restrictions on the knot Floer homologies of K_0 and K_1. One application of this perspective is a simple proof that the map on knot Floer homology induced by a ribbon concordance is injective. Another application is that the U torsion order of knot Floer homology gives a lower bound on the bridge index of a knot. Some of this work is joint (in various pieces) with Adam Levine, Maggie Miller, and Andras Juhasz.