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. This allows us to derive some independence results in the smooth concordance group of knots.

We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology injects in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they've been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. Finally, we present some applications to the study of entanglement in proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

We define a non-standard homolgy theory for ribbon-immersed surfaces in S^3 that captures the homological information of the double-branched cover of the 4-ball with the branch set equal to the push-in of the surface. This is inspired by the Gordon-Litherland description of the homology of the double branched cover over the push-in of an embedded surface. We extend the construction to include general slice surfaces. This is joint work with Brendan Owens.

Ozsvath and Szabo developed an algebraic invariant of a knot that utilises a cut and paste method to determine a bordered invariant whose generators are Kauffman states. The restricted nature of a three strand pretzel knot diagram forces only a small number of possible Kauffman states for these knots, and it makes the determination of this invariant possible using induction on the length of the strands. In this talk, I will give an overview of this construction, an explanation as to why pretzel knots are particularly amenable to study in this way, and the calculation of this invariant (and associated numerical invariants) for families of three strand pretzel knots.

Let $L$ be an $n$-component link with pairwise nonzero linking numbers in a rational homology $3$-sphere $Y$. Assume $X:=Y\setminus\nu(L)$ has nondegenerate Thurston norm and let $S$ be a norm-minimizing surface in $X$. We may obtain a closed surface $\hat{S}$ by Dehn-surgering each component of $L$ according to $\partial S$ and capping off. After some exposition on sutured manifolds, foliations, and some interesting moves one can do on each, I will show that for "most" choices of $[S]$, $\hat{S}$ is norm-minimizing. The proof is constructive; we find a taut foliation in the surgered manifold achieving $\hat{S}$ as a leaf.

This proof leverages Freedman's construction showing that any integer homology sphere bounds a contractible topological 4-manifold and is joint work with JungHwan Park and Peter Teichner.

X.S. Lin defined an invariant of knots in S^3 by counting irreductible SU_2 representations of the knot group with fixed meridinal holonomy. I'll define a similar invariant with SL_2(R) in place of SU_2 and explain why the total Lin invariant (the sum of the two) does not depend on the holonomy parameter. If time permits, I'll give some applications to left orderability of branched covers. This is joint work with Nathan Dunfield.

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 $S^1\times S^2$ 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.

We introduce a new invariant of trisected 4-manifolds, consisting of a set of Spin-C structures. We refer to these as basic classes, given the strong analogy between these classes and basic classes in Donaldson and Seiberg Witten theory. In particular, they satisfy a blow-up formula, the first Chern class of a symplectic structure is a basic class for a Weinstein trisection of a symplectic 4-manifold, and for (g,0)-trisections they satisfy an adjunction inequality. The definition depends on 3-dimensional geometry instead of gauge theory. As in the author's previous work on the Thom conjecture, the adjunction inequality can be established using contact geometry and Khovanov homology. These basic classes are sufficiently strong to resolve the minimal genus problem in rational surfaces. Moreover, they are sufficient to detect exotic smooth structures and show that K3 # -CP^2 is homeomorphic but not diffeomorphic to 3 CP^2 # -20 CP^2. However, the trisection basic classes are not equivalent to Seiberg-Witten basic classes, as they behave differently under finite covers.

The classical 3-genus of knots behaves predictably under the satellite operation: a formula of Schubert states that a winding number $w$ pattern $P$ has $g_3(P(K))= g_3(P)+ |w| g_3(K)$, for an appropriately defined $g_3(P)$ bounded below by $g_3(P(U))$. In this talk, I’ll discuss the extent to which analogous formulae, particularly the intuition that the genus of $P(K)$ should be roughly $|w|$ times the genus of $K$, hold in the context of the slice genus. In the smooth setting the winding number continues to play a pivotal role, but in new joint work with Peter Feller and Juanita Pinz{‘o}n-Caicedo we give surprising evidence that in the topological category the winding number of a pattern is almost irrelevant in its effect on the 4-genus!

Khovanov homology and knot Floer homology are two knot invariants which have many similarities despite significant differences in their constructions. Using an intermediate knot invariant, I will construct a spectral sequence from Khovanov homology to knot Floer homology, proving a conjecture of Rasmussen. I will also describe a local relationship between the invariants which gives a construction of Khovanov homology via a filtration on Ozsváth and Szabó's bordered knot Floer homology. The latter is joint work in preparation with Akram Alishahi.

I will explain how knot Floer homology can be interpreted as a count of closed Reeb orbits in an appropriated contact manifold via the isomorphism between Heegaard Floer homology and embedded contact homology. This is a work in progress with Vincent Colin, Ko Honda and Gilberto Spano.

Given a connected cobordism between two knots in the 3-sphere, we present an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for torus knots. Furthermore, the bridge index of a torus knot is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. Knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and we exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot. This is joint work with Maggie Miller and Ian Zemke.

A link of an isolated complex surface singularity is a 3-manifold that carries a canonical contact structure $\xi_{can}$, given by complex tangencies. Milnor fibers of all possible smoothings (if any) yield Stein fillings of $\xi_{can}$; the resolution of the singularity also gives a filling. An important question is whether all Stein fillings come from the algebraic side: this holds in some simple cases, eg for lens spaces, but fails in general. We will address this question for minimal singularities (a subclass of rational singularities). Deformations and smoothings of minimal singularities are completely understood due to De Jong-Van Straten (1998), in terms of deformations of certain associated plane curve singularities. On the other hand, we showed, jointly with Ghiggini-Golla, that minimal singularities are characterized by $\xi_{can}$ admitting a planar open book. We use Lefschetz fibrations to describe all Stein fillings of these contact structures via a symplectic analog of the De Jong-Van Straten construction, with plane algebraic curves replaced by pseudoholomorphic ones. Then we discuss the differences between the symplectic and algebraic results. (Based on joint work with P. Ghiggini and M. Golla and joint work in progress with L.Starkston. We take a topological approach, no singularity theory background will be assumed.)

We will give knot Floer theoretic obstructions to the existence of rational homology cobordisms from Dehn surgeries on a knot to connected sums of lens spaces (joint work with P.Aceto and J.Park).

Since the early days of mirror symmetry, people have hoped to find Lagrangian torus fibrations on Calabi-Yau 3-folds such that the discriminant locus is a trivalent graph. The problem was that there was no local model for how such a fibration should look in a neighbourhood of a fibre with Euler characteristic -1 (a "negative vertex" of the trivalent graph). In recent work with Mirko Mauri, we construct such a local model, and I think I can explain the construction in 15 minutes.

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of (bordered) Heegaard Floer homology by Lipshitz, Ozsváth and Thurston. In this talk, I shall aim to explain how Koszul duality links previous ‘gluing theorems’ in Heegaard Floer homology (e.g. the mapping cone theorem of Ozsváth and Szabo) with the bordered Heegaard Floer homology of a three-manifold with torus boundary and its ‘pairing theorem’.

We provide a symplectic model for Stroppel’s `extended arc algebras' as Fukaya-Seidel categories of natural Lefschetz fibrations on (n,n)-Springer fibres. This gives a new viewpoint on the spectral sequence from annular to ordinary Khovanov homology. The talk reports on joint work with Cheuk Yu Mak.

I will describe joint work with Andrew Blumberg on the Arnold conjecture for fixed points of Hamiltonian diffeomorphisms. I focus on the key technical point of our work, which uses tools of stable homotopy to establish a version of Poincaré duality for orbifolds without working in characteristic 0.

We give an re-interpretation of Nadler’s arboreal singularities from the perspective of their links, as singular Legendrian spaces. We discuss interpretations from the perspective of constructable sheaf theory, and of loose Legendrians.

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.

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 give rise to non-orientable genus bounds. I’ll try to sketch some of this story.

Utilizing work of Hutchings, Lipshitz and Kutluhan-Lee-Taubes, one can incorporate the genus of holomorphic curves into the Ozsvath-Szabo Heegaard Floer complex. We will describe how, using the Honda-Kazez-Matic description of the Heegaard Floer complex associated to an open book decomposition, this can be used to define a version of the Oszvath-Szabo contact invariant that is more sensitive both to tightness and to Stein cobordisms. This is joint work with Cagatay Kutluhan, Gordana Matic, and Andy Wand.

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. 

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.

Given a knot $K$ in $S^3$, we say that $p/q$ is a characterizing slope if the oriented homeomorphism type of $p/q$-surgery on $K$ is sufficient to uniquely determine the knot $K$. It is known that for a given torus knot all but finitely many non-integer slopes are characterizing and that for hyperbolic knots all but finitely many slopes with $q >2$ are characterizing. I will discuss the proofs of these results, which have a surprising amount in common.

A Floer homology is an invariant of a closed, oriented 3-manifold Y that arises as the homology of a chain complex whose generators are either the set of solutions to a differential equation or the intersection points between Lagrangian manifold, and its differential arises as the count of solutions of a differential equation on Y \times \R. The Instanton Floer chain complex is generated by flat connections on a principal SU(2)-bundle over, and the differential counts solutions to the Yang-Mills equation (known as instantons). The Heegaard Floer chain complex is generated by the intersection points of curves in a Heegaard diagram for Y and its differential counts solutions to the Cauchy-Riemann equation (known as pseudoholomorphic Whitney discs). In the talk I will show that these invariants are the same when the 3-manifold is integral surgery on S^3 along a torus knot. This is joint work with Tye Lidman and Christopher Scaduto.