Tag - Knot theory

Eric Samperton: Topological quantum computation is hyperbolic

12th October, 2023

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the colouring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

Georgios Dimitroglou Rizell: A Relative Calabi-Yau Structure for Legendrian Contact Homology

31st March, 2023

The duality long exact sequence relates linearised Legendrian contact homology and cohomology and was originally constructed by Sabloff in the case of Legendrian knots. We show how the duality long exact sequence can be generalised to a relative Calabi-Yau structure, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficiens in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.

Morgan Weiler: Dynamics of Seifert Surfaces of Torus Knots Via ECH

20th March, 2023

Embedded contact homology (ECH) is a diffeomorphism invariant of three-manifolds due to Hutchings, defined using a contact form. This very diffeomorphism invariance makes it quite useful when studying contact dynamics, because it is possible to apply calculations using simpler contact forms to situations involving more complex ones. We will outline how a knot filtration on ECH was used in a 2015 paper of Hutchings to identify low mean action periodic orbits of disk maps, as well as several more recent generalizations due to other authors. We will then explain the correspondence between the existence of an action function and the construction of a contact three-manifold (via a mapping torus) when starting with a specific surface symplectomorphism. Finally, we will mention how the ECH computations change by analyzing the case of T(2,3) and its genus one Seifert surface. Based on work in progress with Jo Nelson. Note: The material discussed in this talk will differ from that of Jo Nelson's February 28 talk. However, enough background will be given to make this talk self-contained.

Jo Nelson: From Embedded Contact Homology to Surface Dynamics

27th February, 2023

I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of S3 along T(2,q) torus knots to deduce information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages which are freely isotopic to rotation by 1/(2q) along the boundary. I will explain the interplay between the topology of the open book, its presentation as an orbi-bundle, and our computation of the knot filtered ECH chain complex. I will describe how knot filtered ECH realizes the relationship between the action and linking of Reeb orbits and its application to the study of the Calabi invariant and periodic orbits of symplectomorphisms of the pages.

David White: Symplectic Instanton Homology of Knots and Links in 3-manifolds

10th February, 2023

Powerful homology invariants of knots in 3-manifolds have emerged from both the gauge-theoretic and the symplectic kinds of Floer theory: on the gauge-theoretic side is the instanton knot homology of Kronheimer-Mrowka, and on the symplectic the (Heegaard) knot Floer homology developed independently by Ozsváth-Szabó and by Rasmussen. These theories are conjecturally equivalent, but a precise connection between the gauge-theoretic and symplectic sides here remains to be understood. We describe a construction designed to translate singular instanton knot homology more directly into the symplectic domain, a so-called symplectic instanton knot homology: We define a Lagrangian Floer homology invariant of knots and links which extends a 3-manifold invariant developed by H. Horton. The construction proceeds by using specialized Heegaard diagrams to parametrize an intersection of traceless SU(2) character varieties. The latter is in fact an intersection of Lagrangians in a symplectic manifold, giving rise to a Lagrangian Floer homology. We discuss its relation to singular instanton knot homology, as well as the formal properties which this suggests and methods to prove these properties.

Roger Casals: A Microlocal Invitation to Lagrangian Fillings

11th November, 2022

We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding Legendrian knots in the standard contact 3-sphere. Thanks to results from the microlocal theory of sheaves, which we will survey, we then show that this geometric problem gives rise to an interesting moduli space. In fact, we establish a bridge translating geometric operations, such as Lagrangian disk surgeries, into algebraic properties of this moduli space, such as the existence of cluster algebra structures. The talk is intended for a broad symplectic audience and all key ideas will be introduced and motivated.

Ipsita Datta: Lagrangian cobordisms, enriched knot diagrams, and algebraic invariants

4th November, 2022

We introduce new invariants to the existence of Lagrangian cobordisms in ℝ4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries.

We develop appropriate sign conventions and results to characterize boundary points of 1-dimensional moduli spaces with boundaries on Lagrangian tangles. We then use these to define (SFT-like) algebraic structures that recover the previously described obstructions.

Joel Fine: Knots, minimal surfaces and J-holomorphic curves

29th April, 2022

Let K be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space, H4. The main theme of my talk is that it should be possible to count minimal surfaces in H4 which fill K and obtain a link invariant. In other words, the count doesn’t change under isotopies of K. When one counts minimal disks, this is a theorem. Unfortunately there is currently a gap in the proof for more complicated surfaces. I will explain 'morally' why the result should be true and how I intend to fill the gap. In fact, this (currently conjectural) invariant is a kind of Gromov–Witten invariant, counting J-holomorphic curves in a certain symplectic 6-manifold diffeomorphic to S2×H4. The symplectic structure becomes singular at infinity, in directions transverse to the S2 fibres. These singularities mean that both the Fredholm and compactness theories have fundamentally new features, which I will describe. Finally, there is a whole class of infinite-volume symplectic 6-manifolds which have singularities modelled on the above situation. I will explain how it should be possible to count J-holomorphic curves in these manifolds too, and obtain invariants for links in other 3-manifolds.

Joshua Sabloff: Non-Orientable Lagrangian Fillings of Legendrian Knots

19th April, 2022

Most work on Lagrangian fillings of Legendrian knots to date has concentrated on orientable fillings, but instead I will present some first steps in constructions of and (especially) obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings.

Agniva Roy: Constructions of High-Dimensional Legendrians and Isotopies

25th March, 2022

I will talk about an ongoing project that explores the construction of high-dimensional Legendrian spheres from supporting open books and contact structures. The input is a Lagrangian disk filling of a Legendrian knot in the binding. We try to understand the relationship between different constructions from the same input, and suggest parallels, in the S2n+1 case, to a construction defined by Ekholm for ℝ2n+1.