Schedule for: 19w5118 - Unifying 4-Dimensional Knot Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

An $n$-knot is a locally flat PL $n$-sphere in $S^{n+2}$. We show that many decision problems about $n$-knot groups, $n \ge 3$, and groups of closed orientable surfaces in $S^4$, are unsolvable. We will also discuss the case of 2-knots, where the corresponding questions are still open. This is joint work with Fico González-Acuña and Jonathan Simon.

An overview of diagrammatic and algebraic methods for knotted surfaces will be given. Diagrammatic methods include broken surface diagrams, movies, and their moves. Algebraic aspects include quandles and their cocycle invariants, with focus on close connection between diagrams and algebraic properties.

The crossing number of a knot in 3-space is one of the fundamental invariants. Indeed, the knot table used widely in knot theory is based on the crossing number. In surface-knot theory, we can consider the triple point number of a surface-knot in 4-space. Although the triple point number is analogous to the crossing number, there are many differences between them. In this talk, we would like to give a survey on how to determine the triple point number from the viewpoint of diagrammatic and algebraic approaches.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will explain (older) joint work with Dror Bar-Natan on the relationship between 4-dimensional knot theory and Lie theory. On the knot theory side, this involves finite type invariants of welded knotted objects, in other words, a certain class of 4-dimensional tangles. On the Lie theory side stands the Kashiwara-Vergne problem, a famous problem that was first solved, after nearly 30 years, in 2006 by Alekseev and Meinrenken. I will also aim to describe more recent efforts to connect this work to related results of Alekseev-Kawazumi-Kuno-Naef, who provide a different topological interpretation of the Kashiwara-Vergne problem in terms of homotopy classes of loops on surfaces.

The study of loop braid groups has been widely developed during the last twenty years, in different domains of mathematics and mathematical physics. They have been called with several names such as motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups, untwisted ring groups,...and others! We will give a glance on how this richness of formulations carries open questions in different areas. Then we will focus on a lift of Artin’s representation for braid groups and of Dahm’s homomorphism for loop braid groups, given by an injection of the (extended) loop braid group into the group of auto- morphisms of a triple, composed by a group, an abelian group, and an action of the first on the second one.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Symplectic topology has been behind many advances in the study of the smooth topology of 4-manifolds. 4-manifolds which admit a symplectic structure give a nice subclass of smooth 4-manifolds where additional tools are available, while still exhibiting many of the exotic and topological phenomena occurring in generic smooth 4-manifolds. We will discuss some of the tools and major theorems that symplectic topology brings to the table. Instead of going into full technical detail, we will focus on the topological applications of these tools to understanding 4-manifolds and the surfaces they contain. We hope to discuss: adjunction formula and standard neighborhoods, using pseudoholomorphic curves to prove existence of certain surfaces or foliations by surfaces, positivity of intersections for uniqueness of surfaces and controlling isotopies, and symplectic branched coverings.

An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signature. We will report on our recent progress (joint with Korkmaz), which resolves the question for all fiber and base genera except for about 30 pairs at the time of writing.

In this talk we will define braided surfaces with caps, which generalizes the notion of braided surfaces to nonribbon surfaces in $D^4$. We show that any surface in $D^4$ can be isotoped to a braided surface with caps with only positive branch points, and that this isotopy can be taken rel boundary if the boundary is already a classical closed braid in $S^3$. We will then discuss some applications of these surfaces to constructing Lefschetz fibrations with prescribed boundary open book decompositions and braiding closed surfaces in $\mathbb{CP}^2$.

The light bulb lemma more or less asserts, subject to a certain hypothesis, that if a surface $S$ has a transverse sphere, then a tube linking $S$ can be isotopically pulled across $S$. Is the failure of this certain hypothesis ever interesting? We offer some preliminary thoughts.

I use Dave Gabai's 4D light bulb theorem to prove an analogous statement in the setting of concordance: if two spheres $R$, $R'$ in a 4-manifold $X$ are homotopic and $R$ admits a dual sphere, then $R$ and $R'$ are concordant (modulo a condition on 2-torsion in the fundamental group of $X$).

All (closed oriented) smooth four-manifolds can be constructed as branched covers of the sphere. In this talk, I will consider a specific class of such covers: irregular 3-fold covers with non-smooth branching sets. This construction is particularly well-suited for explicitly constructing simply-connected manifolds. I'll give an overview of recent results in this area and discuss potential applications to understanding smooth structures and handle decompositions of simply-connected four-manifolds.

The disc embedding theorem for simply connected 4-manifolds was proved by Freedman in 1982 and forms the basis for his proofs of the h-cobordism theorem, the Poincare conjecture, the exactness of the surgery sequence, and the classification of simply connected manifolds, all in the topological category and dimension four. The disc embedding theorem for more general 4-manifolds is proved in the book of Freedman and Quinn. However, the geometrically transverse spheres claimed in the outcome of the theorem are not constructed. We close this gap by constructing the desired transverse spheres. We also outline where and why such transverse spheres are necessary. This is a joint project with Mark Powell and Peter Teichner.

We will discuss the knot theory of proper embeddings of the plane and half-open annulus in $\mathbb{R}^4$, focusing on what is arguably the simplest situation that is intrinsically noncompact. While it is still unknown if a smoothly knotted embedding $S^2\to\mathbb{R}^4$ can be topologically unknotted, there are various ways of realizing the analogous phenomenon for planes and annuli. We will show how to construct and distinguish various families of examples. These can exhibit remarkably different behavior, and can sometimes be explicitly drawn with level diagrams.

This represents joint work with Danny Ruberman. We will start with critical level embeddings of a pair of surfaces that are topologically isotopic but not smoothly isotopic. We will see how these surfaces may be used to construct exotic homotopy classes in the diffeomorphism group of some 4-manifolds and in spaces of embeddings. Invariants to detect exotic diffeomorphisms based on parameterized gauge theory will be defined along with a method to evaluate the invariants in some interesting cases. Given time, the talk would speculate on how these invariants could provide information about intersection patterns of surfaces in 4-manifolds tools.

A recent paper of Levine-Lidman gives examples of simply connected 4-manifolds $X$, homotopy equivalent to a 2-sphere, that do not admit PL spines. In other words, there is no PL (not necessarily locally flat) embedded sphere in $X$ that is a strong deformation retract of $X$. I will show that a family of the Levine-Lidman examples admit a topological spine—a locally PL sphere that is a strong deformation retract of $X$. This is joint work with Hee Jung Kim.

Quandle cohomology and quandle extension theory is developed by modifying group cohomology and group extensions theory. In 1999, J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito introduced an invariant by using quandle cocycles. It is helpful to study classical knots and knotted surfaces, and still many knot theorists have been studying quandle cocycle and quandle cocycle invariant. In this talk, we survey quandle cohomology and quandle extension theory, after then we study quandle 2-cocycles in a connection with group 2-cocycles. We also observe some properties of abelian extension of a quandle which is defined by a quandle 2-cocycle. This is a joint work with Y. Bae and J. S. Carter.

A knot-theoretic ternary quasigroup is an algebraic system which equips a ternary operation coming from oriented (surface-)knot diagrams with region labelings. A shadow biquandle is an algebraic system which equips two binary operations and an action coming from oriented (surface-)knot diagrams with semi-arc (or semi-sheet) labelings and region labelings. Note that the region labeling by a shadow biquandle depends on the semi-arc (or semi-sheet) labeling whereas the region labeling by a knot-theoretic ternary quasigroup does not. In this talk, we show that under some condition, knot-theoretic ternary quasigroup theory and shadow biquandle theory are the same: Homology groups are the same; cocycle invariants for oriented surface-knots are the same.

In this talk, we consider surfaces embedded in 4-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary 4-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$ . We consider applications of this result to bridge trisections of surfaces in 4-manifolds and Gluck twist in $S^4$. This project is joint with Mark Hughes and Maggie Miller.

There have been examples provided in the past of connected sums of a knotted sphere with an unknotted projective plane which are isotopic to the unknotted projective plane. It is an open question as to whether the connected sum of an odd twist spun knot and an unknotted projective plane is isotopic to the unknotted projective plane, but in this talk we discuss when the result is known to be true after a trivial internal stabilization.

Paul Melvin proved that 0-concordant 2-knots have diffeomorphic Gluck twists, but until recently there were no known proofs that there is more than one 0-concordance class. Now Sunukjian and Dai-Miller have found many examples using Heegaard Floer technology applied to the Seifert 3-manifolds which the 2-knots bound. In this talk we give another proof using Alexander ideals. The main theorem is that the Alexander ideal induces a homomorphism from the 0-concordance monoid of 2-knots to the ideal class monoid of $\mathbb{Z}[t,t^{-1}]$. A corollary is that any 2-knot with nonprincipal Alexander ideal cannot be 0-slice, and moreover has no inverse in the 0-concordance monoid. This is the first proof that the monoid is not a group, and gives another proof of the existence of infinitely many linearly independent 0-concordance classes. These techniques also apply to higher genus surfaces, where we give the first results on 0-concordance. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.

For smooth knots in the 3 sphere, concordance has a purely geometrical definition as the equivalence generated by genus 0 cobordisms in the sphere times an interval. For virtual knots, the relation is extended by using diagram-based definitions. Both framed and twisted virtual knots have a rigidity imbued by a choice of unit normal vector field to the knot. This talk presents combinatorial definitions for concordance of framed and twisted virtual knots and slice obstructions coming from self-linking numbers.

We will examine two and three fold branched coverings of a few classical knots and knotted surfaces, and demonstrate the method of charts that allows us to find immersed braidings of these in codimension 2. As much as possible, explicit constructions and recipes will be given.

We prove an analogue of the Kauffman-Murasugi-Thistlethwaite theorem for alternating links in surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. The proof holds more generally for links admitting adequate diagrams and the key ingredient is a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. This result extends the first and second Tait conjectures to alternating links in thickened surfaces and also to alternating virtual links.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.