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y
There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what such a guide should look like.
It is quite a risky undertaking because it is all too easy to offend by omission, misrepresentation or other. I have not attempted a complete literature survey and inevitably these notes reflects my personal view, jaundiced as it may often be. My apologies for any offence caused.
I would like to express my warm thanks to Lukas Lewark, Alex Shumakovitch, Liam Watson and Ben Webster.
[-] There are already too many introductory articles on Khovanov homology and certainly another is not needed. On the other hand by now - 15 years after the invention of subject - it is quite easy to get lost after having taken those first few steps. What could be useful is a rough guide to some of the developments over that time and the summer school Quantum Topology at the CIRM in Luminy has provided the ideal opportunity for thinking about what ...
[+] 57M25 ; 57M27
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y
A surface system for a link in $S^3$ is a collection of embedded Seifert surfaces for the components, that are allowed to intersect one another. When do two $n$–component links with the same pairwise linking numbers admit homeomorphic surface systems? It turns out this holds if and only if the link exteriors are bordant over the free abelian group $\mathbb{Z}_n$. In this talk we characterise these geometric conditions in terms of algebraic link invariants in two ways: first the triple linking numbers and then the fundamental groups of the links. This involves a detailed study of the indeterminacy of Milnor's triple linking numbers.
Based on joint work with Chris Davis, Matthias Nagel and Patrick Orson.
[-] A surface system for a link in $S^3$ is a collection of embedded Seifert surfaces for the components, that are allowed to intersect one another. When do two $n$–component links with the same pairwise linking numbers admit homeomorphic surface systems? It turns out this holds if and only if the link exteriors are bordant over the free abelian group $\mathbb{Z}_n$. In this talk we characterise these geometric conditions in terms of algebraic link ...
[+] 57M25 ; 57M27 ; 57N70
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2 y
In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-dimensional knot theory. However, the theory extends to other ambient four-manifolds, and we will pay particular attention to the setting of complex curves in simple complex surfaces, where the theory produces surprisingly satisfying pictures and leads to interesting results about trisections of complex surfaces.
This talk is based on various joint works with Dave Gay, Peter Lambert-Cole, and Alex Zupan.
[-] In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-...
[+] 57Q45 ; 57M25 ; 57M50
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2 y
In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.
[-] In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...
[+] 57M27 ; 57M50 ; 57M25
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y
In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.
[-] In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...
[+] 57M27 ; 57M50 ; 57M25
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y
In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.
[-] In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in ...
[+] 57M27 ; 57M50 ; 57M25
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y
Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot
[-] Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows ...
[+] 37C27 ; 37C15 ; 37C10 ; 57M25
English
