Abstracts
Peter Feller
Title: Squeezed Knots.
Abstract: Squeezed knots are the knots that arise as slices of minimal genus smooth cobordisms between a positive knot and a negative knot.
This talk aims to show the working 3.5-dimensional topologist that this (apriori peculiar) notion is a worthwhile addition to their tool kit.
Philosophically speaking, the perspective of squeezed knots allows to streamline many calculations of knot invariants and can help to avoid pitfalls when looking for interesting knots to test new invariants. Concretely, we have the following complementary facts:
-Many concordance invariants such as tau and s (and its refinements) behave in a special way on squeezed knots.
-The class of squeezed knots subsumes many beloved classes of knots, including positive, negative, quasipositive, quasinegative, alternating, and homogeneous ones.
Based on joint work with L. Lewark and A. Lobb
Kyle Hayden
Title: Khovanov homology and knotted surfaces.
Abstract: Khovanov homology provides a rich set of tools for studying knots and links in 3-space and surfaces in 4-space. I will survey recent developments that use Khovanov homology to distinguish knotted surfaces in the 4-ball. In particular, I will focus on the role Khovanov homology plays in joint work with Kim, Miller, Park, and Sundberg, where we produce pairs of Seifert surfaces in the 3-sphere with the same genus and boundary that do not become isotopic when their interiors are pushed into the 4-ball, answering a question of Livingston.
Gabe Islambouli
Title: Multisections of knotted surfaces in 4-manifolds.
Abstract: A bridge multisection of a surface is a generalization of a bridge trisection into a decomposition with more pieces. In this talk we will show how to produce an efficient bridge 4-section of each smooth isotopy class of a complex curve in S2xS2. Motivated by these examples, we will discuss genus 1 multisections and provide a set of moves relating any two multisections of a fixed smooth closed orientable 4-manifold.
Lisa Lokteva
Title: New families of rational homology 3-spheres bounding rationally acyclic 4-manifolds.
Abstract: Problem 4.5 on Kirby's list of important problems in low-dimensional topology asks which rational homology 3-spheres bound rationally acyclic 4-manifolds. This question cannot be answered in full generality, but there exist partial answers for some families of 3-manifolds, including lens spaces (Lisca, 2007) and integral surgeries on positive torus knots (Aceto, Golla, Larson, Lecuona, 2020). An important method used in these classifications is lattice embeddings, whose non-existence obstructs the existence of a rationally acyclic filling. Determining if a lattice embedding exists for any manifold in a family often amounts to difficult combinatorics.
Using a computer, Aceto, Golla, Larson and Lecuona found a particularly difficult-to-guess family of integral surgeries on iterated torus knots that bound rationally acyclic manifolds. In my recent work, I explain why this family is not that strange, but is in fact simply the intersection between a large, new and natural family of boundaries of rationally acyclic 4-manifolds and the integral surgeries on positive torus knots.
Marco Marengon
Title: Relative genus bounds in indefinite 4-manifolds
Abstract: Given a closed 4-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X − int(B^4), with boundary a given knot K in the 3-sphere. We give several methods to bound the genus of such surfaces in a fixed homology class. Our techniques include adjunction inequalities from Heegaard Floer homology and the Bauer-Furuta invariants, and the 10/8 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disc) in a 4-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. This is joint work with Ciprian Manolescu and Lisa Piccirillo.
Jean Baptiste Meilhan
Title: Concordance invariants of knotted surfaces via 'cut-diagrams'
Abstract: The purpose of this talk is to define a family of concordance invariants of general knotted surfaces in 4-space. The construction is modeled on Milnor link invariants, which are numerical concordance invariants of links in 3-space, extracted from the nilpotent quotients of the link group. Our construction makes use of "cut-diagrams" of knotted surfaces in 4-space, which encode these objects in a simple combinatorial way. Roughly speaking, for a knotted surface obtained as embedding of the abstract surface S, a cut-diagram is a kind of 1-dimensional diagram on S with some labeling. We will provide some applications, comparing this construction to other known concordance invariants. Joint work with Benjamin Audoux and Akira Yasuhara.
Brendan Owens
Title: Extremal ribbon cobordisms between alternating links and their double branched covers.
Abstract: A ribbon cobordism between links in the 3-sphere is an embedded surface bounded by them in $S^3\times [0,1]$ on which the restriction of the coordinate from the second factor has no maxima. A ribbon cobordism between two 3-manifolds is a cobordism with no 3-handles. Examples of the latter arise from the former by taking the double branched cover.
We consider ribbon cobordisms with Euler characteristic 0 between nonsplit alternating links $L$, and also ribbon rational homology cobordisms between their double branched covers $\Sigma(L)$. We show that the ratio $\det L/4^n$ is preserved or decreased by such cobordisms, where $n+1$ is the number of white regions of a reduced chessboard-coloured alternating diagram. Using Heegaard Floer homology and Donaldson’s diagonalisation theorem, we give a complete criterion for existence of such a cobordism between the double branched covers of alternating links for which this quantity is preserved. For alternating links with $\det L= 4^n$, we show that $\Sigma(L)$ bounds a rational homology ball if and only if $L$ bounds a ribbon surface with Euler characteristic 1 in the 4-ball, and we give a nice classification of these links.
This proves the extremal case of a conjectured complete criterion for when the double branched cover of an alternating link bounds a rational homology ball.
This is joint work with Josh Greene.
Andrea Parma
Title: An introduction to horizontal decompositions.
Abstract: The goal of the talk is to describe a special kind of handle decompositions of 4-dimensional smooth cobordisms. As a motivating example, we will start by showing how they arise in the construction of smooth embeddings of some rational homology balls in $\mathbb{CP}^2$. Subsequently, we will give the general definition of horizontal decompositions and explore their main properties, with focus on the description of horizontality-preserving handle slides, on the existence of such decompositions for generic cobordisms and on some potential applications.
Juanita Pinzon-Caicedo
Title: Satellite Operations that are not homomorphisms.
Abstract: Two knots $K_0$ and $K_1$ are said to be smoothly concordant if the connected sum $K_0\#m(K_1^r)$ bounds a disk smoothly embedded in the 4-ball. Smooth concordance is an equivalence relation, and the set $\mathcal{C}$ of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. Satellite operations, or the process of tying a given knot P along another knot K to produce a third knot P(K), are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern P that suffice to conclude that the function $P:\mathcal{C}\to \mathcal{C}$ is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.
Mark Powell
Title: Spines of homotopy 2-spheres.
Abstract: I will explain how to classify embeddings of the 2-sphere into a 4-manifold, provided the embedding is a homotopy equivalence, and the
fundamental group of the complement is abelian. Joint work with Patrick Orson.
Aby Thompson
Title: Kirby diagrams and trisections
Abstract: I’lll discuss the relation between a Kirby link diagram for a 4-manifold and a trisection diagram, and how one can inform the other, with examples.
Rafael Torres
Title: Examples of nullhomotopic 2-spheres in closed 4-manifolds that are smoothly knotted and topologically unknotted.
Abstract: For a closed simply connected 4-manifold X that satisfies certain conditions, we describe a construction of infinite sets of pairwise smoothly inequivalent nullhomologous 2-spheres that topologically bound a handlebody inside the connected sum of $X#S^2\times S^2$.
Paula Truöl
Title: Strongly quasipositive knots are concordant to infinitely many strongly quasipositive knots
Abstract: We show that every strongly quasipositive knot is smoothly concordant to infinitely many pairwise non-isotopic, strongly quasipositive knots. In contrast to our result, Baader, Dehornoy and Liechti showed that every (topologically locally-flat) concordance class contains at most finitely many positive knots. Moreover, it was conjectured by Baker that smoothly concordant strongly quasipositive and fibered knots are isotopic. Our construction uses a satellite operation with companion a slice knot with maximal Thurston-Bennequin number -1.
Short talks:
Vincent Florens. Title: Seifert forms and slice Euler characteristic. Abstract: We define the Witt coindex of a link with non-trivial Alexander polynomial, as a concordance invariant from the Seifert form. We show that it provides an upper bound for the (locally flat) slice Euler caracteristic of the link, extending the work of Levine on algebraically slice knots and Taylor on the genera of knots. Work in progress with S.Orevkov.
Hyeonhee Jin. Title: Kirby diagrams for Non-orientable 4 manifolds. Abstract: In comparison to the wide use of Kirby diagrams for orientable 4 manifolds, the analogue for non orientable 4 manifolds has been somewhat less common. Here I would follow Akbulut's way of drawing non-orientable one handles and use this to show some diffeomorphisms of sphere bundles over RP2 connected sum with one CP2.
Daniel Kasprowski. Title: Embedding surfaces in 4-manifolds. Abstract: Freedman and Quinn proved a (topological) embedding theorem for spheres in 4-manifolds with good fundamental group in the presence of dual spheres. In joint work with Mark Powell, Arunima Ray and Peter Teichner, we have generalized this to an embedding theorem for surfaces. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a combinatorial formula for its computation. For this we introduce the notion of band characteristic surfaces.
Marc Kegel. Title: Open books on 4-manifolds. Abstract: An open book on a 4-manifold is an embedded surface B together with a fibration of its complement over the circle that admits a standard form near B. This generalizes the notion of a fibered link in a 3-manifold. Due to their close connections to contact geometry open books are mostly studied in odd dimensions. However, the definition of an open book makes sense in all dimensions, in particular in dimension 4.
In this talk, I will first survey what is known about open books in dimensions different from 4. Then I will restrict to dimension 4 and explain what is known and what is not known about open books and their relations to other structures in dimension 4. This is based on joint work in progress with Felix Schmäschke.
Laura Marino. Title: Rational tangles and a new bound for the unknotting number. Abstract: We use a universal version of Khovanov homology to extract a new combinatorial knot invariant l. While l gives a lower bound for the unknotting number, in some cases stronger than other known bounds, in fact more is true: it is a lower bound for the proper rational unknotting number, the minimal number of rational tangle replacements relating a knot to the unknot. This is based on joint work with D. Iltgen and L. Lewark.
Arunima Ray. Title: In search of counterexamples. Abstract: There are a variety of equivalence relations on 4-manifolds, such as h-cobordism, homotopy equivalence, homeomorphism, and diffeomorphism. I will share a flowchart describing the relationships between them, and the main remaining open questions. I will then attempt to recruit the attendees of the conference to help compile a similar flowchart and zoo of counterexamples for surfaces in 4-manifolds, according to equivalence relations such as stable isotopy, concordance, and homotopy. The goal of the endeavour would be to write a joint survey paper, and to uncover remaining open questions.