London Low Dimensional Topology Seminar

15th October: Steven Sivek, Imperial College London

Title: Ribbon concordance and fibered predecessors 

Abstract: Ribbon concordance defines an interesting relation on knots.  In his initial work on the topic, Gordon asked whether it is a partial order, and this question was open for over 40 years until Agol answered it affirmatively in 2022.  However, we still don’t know many basic facts about this partial order: for example, does any infinite chain of ribbon concordances  eventually stabilize?  Even better, if we fix a knot  in the 3-sphere, are there only finitely many knots that are ribbon concordant to ?  I’ll talk about joint work with John Baldwin toward these questions, in which we use tools from both Heegaard Floer homology and hyperbolic geometry to say that at the very least, there are only finitely many fibered hyperbolic knots ribbon concordant to. 

22nd October: Misha Schmalian, Oxford University

Title: Uniqueness of Dehn Surgery 

Abstract: Dehn Surgery is an operation that allows us to describe any three-manifold via a framed link in the three-sphere. How unique is such a description? Recently, there has been substantial progress and attention on questions of this nature. In this talk I will: 

1. Give an introduction to Dehn surgery problems and how tackle them using Geometrization;

2. Discuss some recent and some upcoming work with Marc Kegel on this topic; 

3. Give a longer exposition of open questions in the area (not all of which look impossible).  

29th October: Maartje Wisse, LSGNT/King's College London

Title: The Dowlin Spectral Sequence for Dummies

Abstract: Rasmussen's 2005 conjecture that there should be a spectral sequence from Khovanov Homology to Knot Floer Homology was fully proved by Dowlin in 2018, following on from various similar earlier results. In this talk I will cover the motivation to build such an object, as well as an outline of the arguments used in Dowlin's construction. If time allows I will lay out the work I am doing with this. This is intended to be an overview talk and will be as light on the details as is possible!

5th November: No seminar for KCL Reading Week!

12th November: Gheehyun Nahm, Princeton University

Title: Barbells and Knotted things

Abstract:  In 2019, Budney and Gabai defined barbell diffeomorphisms and used them to construct knotted 3-balls in S^4. Barbell diffeomorphisms provide a concise and hands-on method to produce interesting diffeomorphisms of 4-manifolds. After introducing barbell diffeomorphisms, we explore further applications of them: first, we construct knotted S^3’s in S^5 with four critical points with respect to the standard height function. This, in particular, produces knotted solid tori in S^4, proving the remaining case of a conjecture of Budney and Gabai. Time permitting, we also use barbell diffeomorphisms to construct Brunnian links of 3-balls in S^4. This is joint work with Seungwon Kim and Alison Tatsuoka. 

19th November: Selim Ghazouani, UCL

Title: Dynamical systems, surface homeomorphisms and 3-manifolds          

Abstract:  I will try to explain how Thurston's introduction of dynamical systems in low-dimensional topology revolutionised the field in the 70s. 

26th November: Soheil Azarpendar, University of Oxford

Title: On Fox’s trapezoidal conjecture

Abstract: An old and challenging conjecture proposed by R. H. Fox in 1962 states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot are trapezoidal, i.e. they strictly increase, possibly plateau, then strictly decrease. In this talk, I will describe a recent development on the conjecture, based on joint work with András Juhász and Tamás Kálmán, where we prove it for diagrammatic plumbings of special alternating links and obtain partial results for alternating three-braid closures.

3rd December: Alessandro Cigna, King's College London

Title: Constructing Taut Foliations: Ideas from Gabai’s Theory

Abstract: Taut foliations provide powerful insight into the topology of 3-manifolds, yet determining which manifolds admit such structures remains a central question in low-dimensional topology. In 1980, Gabai introduced the framework of sutured manifolds, offering a systematic way to construct taut foliations and to analyse their behaviour under topological operations. Using this machinery, he proved that every oriented, compact, irreducible 3-manifold with torus boundary and positive first Betti number admits a taut foliation. He also applied similar techniques to knot theory, showing that the Seifert algorithm produces minimal-genus Seifert surfaces for alternating links.
In this talk, I will introduce the sutured manifold theory developed by Gabai and outline the ideas behind these influential results.

10th December: Abigail Hollingsworth, University of Warwick

Title: From the boundary of Dehn surgery space to the real locus of the shape variety.

Abstract: The complement of the 5_2 knot is a hyperbolic three-manifold. We will take its ideal triangulation and define its shape variety from the edge gluings of the triangulation. Defining the Dehn filling equation for three-manifolds, we will build the Dehn surgery space of the 5_2 knot complement. 

In many pictures of Dehn surgery space, we see the real representations corresponding to straight lines on the boundary of Dehn surgery space. Is this always true?

A 'nice' type of flattening of tetrahedra shapes is a called a taut flattening. Given a path of positive volume tetrahedra to a flattening, that flattening is always taut.

Is it possible for a taut flattening to be approached by a path of tetrahedra that contain some negative volume tetrahedra?

What other questions can we ask about the relationships between real shapes on the shape variety and the boundary of Dehn surgery space?