# Seminars & Events for Topology Seminar

##### Twisted Alexander polynomials and knot concordance

Fox and Milnor showed that the classical Alexander polynomial of a knot K in S^3 can obstruct K from bounding a smooth disk in B ^4 . Twisted Alexander polynomials provides further information about four-dimensional aspects of classical knots. This information is most easily formulated in terms of the knot concordance group. This talk will be a survey of some of the uses of the twisted polynomial in studying properties of knot concordance. The talk will include a discussion of the affects of knot reversal and mutation on concordance, the classification of low-crossing number knots in the concordance group, and doubly slicing knots. The results represent joint work with Julia Collins, Paul Kirk, and Jeff Meier.

##### Taut foliations on graph manifolds

An L-space is a rational homology sphere with simplest possible Heegaard Floer homology. Ozsváth and Szabó have shown that if a closed, connected, orientable three-manifold has a coorientable taut foliation then it is not an L-space. I will explain how to prove the converse to this statement when restricting to graph manifolds. Combined with work of Boyer and Clay, this leads to an equivalence between graph manifold L-spaces and graph manifolds with non-left-orderable fundamental group. This is joint work with J. Hanselman, J. Rasmussen, and S. Rasmussen.

##### Casson towers and slice knots

A link is slice if it is the boundary of a disjoint union of flat discs in the 4-ball. The link slicing problem is closely related to the surgery and s-cobordism programme for classifying 4-manifolds. A Casson tower is a 4-manifold with boundary built from iteratively attempting to find an embedded disc in a 4-manifold using immersed discs. The number of iterated attempts is called the height of the Casson tower. I will describe results on obtaining embedded discs from Casson towers of height 4, 3 and 2, from joint work with Jae Choon Cha. The height 2 result in particular enabled us to construct an interesting class of new slice knots.

##### Quantum invariants in the (oriented) knot Floer cube of resolutions

Using filtrations on the knot Floer cube of resolutions, I will define a triply graded invariant that categorifies the HOMFLY-PT polynomial and has a spectral sequence converging to HFK. Along the way I will discuss relationships between this construction and deformations of sl(n) homology.

##### Heegaard Floer homology for tangles and cobordisms between them

Heegaard Floer homology was generalized for non-closed 3-manifolds with certain boundary decoration called sutured manifolds, by Juhasz, Eftekhary and I. Sutured manifolds can be described as a generalization of oriented tangles. We use this description to define a notion of cobordism between sutured manifolds. Associated with cobordisms between sutured manifolds we will describe the definition of an invariant homomorphism between Heegaard Floer homologies of the corresponding sutured manifolds. These maps generalize cobordism maps associated to 4-dimensional cobordisms between closed 3-manifolds and define cobordism maps for decorated cobordisms between pointed knots. This is a joint work with Eaman Eftekhary.

##### A tale of two norms

The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms. Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below. Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth in homology of towers of finite covers. After carefully introducing both norms, I will discuss new results that refine and clarify the precise relationship between them; one tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock.

##### The Seiberg-Witten equations and knots in 3-manifolds

Given a null-homologous knot inside a closed oriented 3-manifold, I will describe a filtered chain complex using the Seiberg-Witten equations, and discuss its relationship with homological invariants of knots defined by Ozsvath-Szabo and Kronheimer-Mrowka. This is work in preparation.

##### Seifert pairing bounds on the topological slice genus of knots

We present upper bounds for the topological slice genus of knots coming from the Seifert pairing. The main ingredient in the proof is 3-dimensional reduction to a consequence of Freedman's disk theorem: knots with trivial Alexander polynomial are topologically slice. These bounds yield surprising consequences for simple families of knots such as torus knots and 2-bridge knots that contrast smooth results coming from the Thom conjecture and Donaldson's diagonalization theorem. The talk is based on joint work with McCoy and Baader, Lewark, and Liechti.

##### TBA - Introduction; Nick Sheridan

**This is a joint Geometry/Topology day.** There will be a short introduction of the RTG event, followed by Nicholas Sheridan's talk at 12:10.

##### Hyperbolic 3-manifolds with low cusp volume

**This is a joint Geometry/Topology day. **The past fifteen years have seen a great deal of progress towards a complete picture of hyperbolic manifolds of low volume. The volume of a hyperbolic manifold is a topological invariant and can be viewed as a measure of complexity. In fact, there are only finitely many hyperbolic manifolds of a given volume. For hyperbolic 3-manifolds with cusps, one can also consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results and techniques for understanding the infinite families of hyperbolic 3-manifolds of low cusp volume. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings.

##### TBA - Marco Aurelio Mendez Guaraco

**This is a joint Geometry/Topology day.**

##### Knot Concordance Invariants and Heegaard Floer Homology

**This is a joint Geometry/Topology day. ** I will define knot concordance and discuss a set of concordance invariants that are constructed using Heegaard Floer homology.

##### Min-max theory and least area minimal hypersurfaces

**This is a joint Geometry/Topology day. ** Min-max theory is a powerful way for constructing minimal hypersurfaces and has numerous geometric applications. In this talk, I will present one of them due to Calabi and Cao: on a convex sphere, any closed geodesic of least length is simple. I will explain how to extend this result to higher dimensions.

##### Cube of resolutions complexes for Khovanov-Rozansky homology and knot Floer homology

**This is a joint Geometry/Topology day. **I will compare the oriented cube of resolutions constructions for Khovanov-Rozansky homology and knot Floer homology. Manolescu conjectured that for singular diagrams (or trivalent graphs) the HOMFLY-PT homology and knot Floer homology are isomorphic - I will show that this conjecture is equivalent to a certain spectral sequence collapsing. This will also lead to a recursion formula for the HOMFLY-PT homology of singular diagrams that categorifies Jaeger's composition product formula.

##### Bridge trisections of knotted surfaces in the four-sphere

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links - every such surface admits a decomposition into three standard pieces called a bridge trisection. I'll describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface. This is joint work with Alexander Zupan.

##### Some properties of Pin(2)-monopole Floer homology

In this talk I will discuss the basic properties and examples of Pin(2)-monopole Floer homology (including some simple computational tools). This is the Morse-theoretic analogue of Manolescu's Pin(2)-equivariant Seiberg-Witten-Floer homology, and it can be used to provide an alternative disproof of the longstanding Triangulation Conjecture.

##### Knot contact homology and string topology

A natural question about knot contact homology, a knot invariant with origins in contact geometry, is what information it contains about the topology of a knot. Until recently we had some understanding of this, but the understanding was rather ad hoc. I will discuss a new way to describe a key part of knot contact homology, the "cord algebra", through string topology. This allows us to interpret the cord algebra in terms of the fundamental group of the knot complement, and in particular to conclude that knot contact homology detects the unknot and (by work of Gordon and Lidman) torus knots. This is joint work with Kai Cieliebak, Tobias Ekholm, and Janko Latschev.

##### Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, $\pi_1$-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

##### Quasigeodesic Pseudo-Anosov flows in hyperbolic 3-manifolds

We obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover. We prove that such flows are quasigeodesic if and only if there is an upper bound, depending only on the flow, to the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow.