In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite link is one that orbits a companion knot K in the sense that it lies inside a regular neighborhood of the companion. A satellite knot can be picturesquely described as follows: start by taking a nontrivial knot lying inside an unknotted solid torus . Here "nontrivial" means that the knot is not allowed to sit inside of a 3-ball in and is not allowed to be isotopic to the central core curve of the solid torus. Then tie up the solid torus into a nontrivial knot. This means there is a non-trivial embedding and . The central core curve of the solid torus is sent to a knot , which is called the "companion knot" and is thought of as the planet around which the "satellite knot" orbits. The construction ensures that is a non-boundary parallel incompressible torus in the complement of . Composite knots contain a certain kind of incompressible torus called a swallow-follow torus, which can be visualized as swallowing one summand and following another summand. Since is an unknotted solid torus, is a tubular neighbourhood of an unknot . The 2-component link together with the embedding is called the pattern associated to the satellite operation. A convention: people usually demand that the embedding is untwisted in the sense that must send the standard longitude of to the standard longitude of . Said another way, given any two disjoint curves , preserves their linking numbers i.e.: . (Wikipedia).
Satellite operations and Legendrian knot theory - John Etnyre
Satellite operations and Legendrian knot theory Augmentations and Legendrians at the IAS Topic: Satellite operations and Legendrian knot theory Speaker: John Etnyre Date: Thursday, February 11 Satellite operations are a common way to create interesting knot types in the smooth category. I
From playlist Mathematics
Link: https://www.geogebra.org/m/a72HSgzU
From playlist Geometry: Challenge Problems
This knot is very useful for adjusting tie downs quickly and easily. For example, a tarp could be held down by a series of these knots and be made very tight so the wind cannot make it rise, and easily be removed simply by sliding the knots later. A taut line knot is also used to keep larg
From playlist Practical Projects & Skills
Kimihiko Motegi: L-space knots in twist families and satellite L-space knots
Abstract: Twisting a knot K in S3 along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. Comparing the behaviors of the Seifert genus g(Kn) and the slice genus g4(Kn) under twistings, we prove that if g(Kn)−g4(Kn) [is less than] C for some constant C for i
From playlist Topology
Link: https://www.geogebra.org/m/JEk3MHvc
From playlist Geometry: Challenge Problems
Algebraic topology: Fundamental group of a knot
This lecture is part of an online course on algebraic topology. We calculate the fundamental group of (the complement of) a knot, and give a couple of examples. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52yxQGxQoxwOtjIEtxE2BWx
From playlist Algebraic topology
This step by step guide demonstrates tying 15 types: 00:36 Overhand 01:22 Square 02:36 Figure Eight 03:40 Bowline, 05:29 Running 06:19 Half, 07:45 Timber, 09:42 Rolling, 10:43 Clove Hitches 11:30 Cat's Paw 12:58 Single, 14:40 Double Sheet or Becket Bends 15:30 Fisherman's, 17:09 Doubl
From playlist How To Tutorials
Link: https://www.geogebra.org/m/bd69d6u4
From playlist Geometry: Challenge Problems
Colin Adams: Hyperbolic Volumes of Virtual Knots and their Compositions
Colin Adams, Williams College Title: Hyperbolic Volumes of Virtual Knots and their Compositions Hyperbolic Volumes of Virtual Knots and their Compositions\\ \noindent\textbf{Abstract:} Many knots are known to be hyperbolic and therefore have a hyperbolic volume. But composite knots are n
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Jessica Purcell - Lecture 1 - Hyperbolic knots and alternating knots
Jessica Purcell, Monash University Title: Hyperbolic knots and alternating knots Hyperbolic geometry has been used since around the mid-1970s to study knot theory, but it can be difficult to relate geometry of knots to a diagram of a knot. However, many results from the 1980s and beyond s
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Taut foliations and cyclic branched covers - Cameron Gordon
Cameron Gordon, Univ Texas Workshop on Flows, Foliations and Contact Structures 2015-2016 Monday, December 7, 2015 - 08:00 to Friday, December 11, 2015 - 12:00 This workshop is part of the topical program "Geometric Structures on 3-Manifolds" which will take place during the 2015-2016 aca
From playlist Workshop on Flows, Foliations and Contact Structures
Charles Stine: The Complexity of Shake Slice Knots
Charles Stine, Brandeis University Title: The Complexity of Shake Slice Knots It is a well studied conjecture that a shake slice knot is in fact slice. Many counterexamples have been given, but most are close to being slice in a technical sense. In this talk, we will give a precise way to
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Link: https://www.geogebra.org/m/cjS3b6Zr
From playlist Geometry: Challenge Problems
Sergey Melikhov - Not all links are isotopic to PL links
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Sergey Melikhov, Steklov Math Institute (Moscow) Title: Not all links are isotopic to PL links Abstract: Two links in the 3-sphere are called (non-ambiently) isotopic if they are homotopic through embeddings. D. Rolfsen (
From playlist 38th Annual Geometric Topology Workshop (Online), June 15-17, 2021
Knot Theory and Machine Learning - Andras Juhasz
DeepMind Workshop Topic: Knot Theory and Machine Learning Speaker: Andras Juhasz Affiliation: University of Oxford Date: March 28, 2022 The signature of a knot K in the 3-sphere is a classical invariant that gives a lower bound on the genera of compact oriented surfaces in the 4-ball wi
From playlist DeepMind Workshop
Ocean Discovery Talk | Whale Rescue
How do first responders rescue whales when they're tangled in fishing gear or marine debris? Join Exploratorium staff member and NOAA-trained entanglement responder Kathi George to learn how responders approach and save entangled whales. These majestic animals weigh 50,000 pounds, and resp
From playlist Earth Science
MIT 16.687 Private Pilot Ground School, IAP 2019 Instructor: Philip Greenspun, Tina Srivastava View the complete course: https://ocw.mit.edu/16-687IAP19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63cUdAG3v311Vl72ozOiK25 This lecture focused on how to navigate an ai
From playlist MIT 16.687 Private Pilot Ground School, IAP 2019
Intersection of Planes on Geogebra
In this video, we look at a strategy for finding the intersection of planes on Geogebra.
From playlist Geogebra