Knot invariants | Alternating knots and links
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating prime knots have 8 crossings (and there are three such: 819, 820, 821). It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in knot theory and 3-manifold theory, due to their complements having useful and interesting geometric and topological properties. This led Ralph Fox to ask, "What is an alternating knot?" By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots. In November 2015, Joshua Evan Greene published a preprint that established a characterization of alternating links in terms of definite spanning surfaces, i.e. a definition of alternating links (of which alternating knots are a special case) without using the concept of a link diagram. Various geometric and topological information is revealed in an alternating diagram. Primeness and splittability of a link is easily seen from the diagram. The crossing number of a , alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait conjectures. An alternating knot diagram is in one-to-one correspondence with a planar graph. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner. (Wikipedia).
What is the alternate in sign sequence
👉 Learn about sequences. A sequence is a list of numbers/values exhibiting a defined pattern. A number/value in a sequence is called a term of the sequence. There are many types of sequence, among which are: arithmetic and geometric sequence. An arithmetic sequence is a sequence in which
From playlist Sequences
What is an alternating series? - Week 4 - Lecture 5 - Sequences and Series
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From playlist Ohio State: Calculus Two with Jim Fowler: Sequences and Series | CosmoLearning Mathematics
This calculus 2 video tutorial provides a basic introduction into the alternating series test and how to use it to determine the convergence and divergence of a series. You need to show that the sequence goes to zero as n goes to infinity and you need to establish that the sequence is dec
From playlist New Calculus Video Playlist
Angles involving Parallel Lines
"Recognise vertically opposite, alternate, corresponding and cointerior angles."
From playlist Shape: Angles
This video explains how to apply the alternating series test. http://mathispower4u.yolasite.com/
From playlist Infinite Sequences and Series
Alternating Series | Definition and Convergence
Learning Objectives: 1) Define an Alternating Series 2) State the conditions for an Alternating Series to be convergent 3) Understand the intuition behind the Alternating Series Test 4) Apply the Alternating Series Test to specific examples, verifying all assumptions This video is part o
From playlist Older Calculus II (New Playlist For Spring 2019)
The alternating series. Test for convergence.
From playlist Advanced Calculus / Multivariable Calculus
Calculus 11.5 Alternating Series
My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart
From playlist Calculus
Ex: Apply Alternating Series to Infinite Series - Divergent
This video provides an example of applying the alternating series test to a divergent alternating series. Site: http://mathispower4u.com
From playlist Infinite Series
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
Hyperbolic Knot Theory (Lecture - 2) by Abhijit Champanerkar
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)
Nafaa Chbili: On Quasi-alternating Links
Nafaa Chbili, United Arab Emirates University Title: On Quasi-alternating Links An interesting class of knots and links has been introduced by Ozsv{\'a}th and Szab{\'o} while studying the Heegaard Floer homology of the branched double-covers of alternating links. The homological properties
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
Jessica Purcell: Structure of hyperbolic manifolds - Lecture 3
Abstract: 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
From playlist Topology
András Stipsicz - Upsilon invariants of knots
June 22, 2018 - This talk was part of the 2018 RTG mini-conference Low-dimensional topology and its interactions with symplectic geometry
From playlist 2018 RTG mini-conference on low-dimensional topology and its interactions with symplectic geometry I
David BROADHURST - Tasmanian Adventures
I report on two adventures with Dirk Kreimer in Tasmania, 25 years ago. One of these, concerning knots, is not even wrong. The other, concerning a conjectural 4-term relation, is either wrong or right. I suggest that younger colleagues have powerful tools that might be brought to bear on t
From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
Khovanov Homology and Virtual Knot Cobordism by Louis H. Kauffman
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)
Determining clockwise vs counter clockwise rotations
👉 Learn how to rotate a figure and different points about a fixed point. Most often that point or rotation will be the original but it is important to understand that it does not always have to be at the origin. When rotating it is also important to understand the direction that you will
From playlist Transformations
Mathematical Hugs (and Chiral Knots) - Numberphile
Extra footage at: https://youtu.be/ue9LHv4XXBQ - Featuring Ayliean MacDonald. More links & stuff in full description below ↓↓↓ More about Ayliean MacDonald: https://linktr.ee/Ayliean Ayliean videos on Numberphile: https://bit.ly/Ayliean_Playlist Knots Playlist: http://bit.ly/Knot-a-Phil
From playlist Ayliean MacDonald on Numberphile