In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal". A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant. Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same. From the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves ("triangular moves"). Tricolorability (and n-colorability) is a particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology. Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot. Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants). The complement of a knot itself (as a topological space) is known to be a "complete invariant" of the knot by the Gordon–Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The peripheral subgroup can also work as a complete invariant. By Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation. In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants. There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fáry–Milnor theorem states that if the total curvature of a knot K in satisfies where κ(p) is the curvature at p, then K is an unknot. Therefore, for knotted curves, An example of a "physical" invariant is ropelength, which is the length of unit-diameter rope needed to realize a particular knot type. (Wikipedia).
Clément Maria (10/23/19): Parameterized complexity of quantum invariants of knots
Title: Parameterized complexity of quantum invariants of knots Abstract: We give a general fixed parameter tractable algorithm to compute quantum invariants of knots presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the knot diagram.
From playlist AATRN 2019
Untangling the beautiful math of KNOTS
Visit ► https://brilliant.org/TreforBazett/ to help you learn STEM topics for free, and the first 200 people will get 20% off an annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor Suppose yo
From playlist Cool Math Series
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
Knots and surfaces II | Algebraic Topology | NJ Wildberger
In the 1930's H. Siefert showed that any knot can be viewed as the boundary of an orientable surface with boundary, and gave a relatively simple procedure for explicitly constructing such `Seifert surfaces'. We show the algorithm, exhibit it for the trefoil and the square knot, and then di
From playlist Algebraic Topology
Link: https://www.geogebra.org/m/JEk3MHvc
From playlist Geometry: Challenge Problems
Link: https://www.geogebra.org/m/a72HSgzU
From playlist Geometry: Challenge Problems
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
Knots, Virtual Knots 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)
Knots and surfaces I | Algebraic Topology | NJ Wildberger
This lecture is an introduction to knot theory. We discuss the origins of the subject, show a few simple knots, talk about the Reidemeister moves, and then some basic invariants, namely minimal crossing number, linking number (for links) and then the Alexander-Conway polynomial. This is p
From playlist Algebraic Topology
Topologically Ordered Matter and Why You Should be Interested by Steven H. Simon
COLLOQUIUM TOPOLOGICALLY ORDERED MATTER AND WHY YOU SHOULD BE INTERESTED SPEAKER: Steven H. Simon (Oxford University, United Kingdom) DATE: Mon, 26 October 2020, 15:30 to 17:00 VENUE: Online ABSTRACT In two dimensional topologically ordered matter, processes depend on gross topology
From playlist ICTS Colloquia
Marc Lackenby - Using machine learning to formulate mathematical conjectures - IPAM at UCLA
Recorded 14 February 2023. Marc Lackenby of the University of Oxford presents "Using machine learning to formulate mathematical conjectures" at IPAM's Machine Assisted Proofs Workshop. Abstract: I will describe how machine learning can be used as a tool for pure mathematicians to formulate
From playlist 2023 Machine Assisted Proofs Workshop
Rima Chatterjee: Structure Theorems of Legendrian Knots in Contact Manifolds
Rima Chatterjee, University of Cologne Title: Structure Theorems of Legendrian Knots in Contact Manifolds Structure theorems of a Legendrian knot have been an interesting study in contact geometry. One can ask when topological operations on a knot gives us important information about the g
From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022
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
Three Knot-Theoretic Perspectives on Algebra - Zsuzsanna Dancso
Zsuzsanna Dancso University of Toronto; Institute for Advanced Study September 21, 2011 For more videos, visit http://video.ias.edu
From playlist Mathematics
The Signature and Natural Slope of Hyperbolic Knots - Marc Lackenby
DeepMind Workshop Topic: The Signature and Natural Slope of Hyperbolic Knots Speaker: Marc Lackenby Affiliation: University of Oxford Date: March 30, 2022 Andras Juhasz has explained in his talk how machine learning was used to discover a previously unknown relationship between invariant
From playlist DeepMind Workshop
Knot polynomials from Chern-Simons field theory and their string theoretic... by P. Ramadevi
Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries
From playlist Quantum Fields, Geometry and Representation Theory
Cabling of knots in overtwisted contact manifolds - Rima Chatterjee
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Title: Cabling of knots in overtwisted contact manifolds Speaker: Rima Chatterjee Affiliation: Cologne Date: October 8, 2021 Abstract: Knots associated to overtwisted manifolds are less explored. There are two types of kno
From playlist Mathematics
Symmetries show up everywhere in physics. But what is a symmetry? While the symmetries of shapes can be interesting, a lot of times, we are more interested in symmetries of space or symmetries of spacetime. To describe these, we need to build "invariants" which give a mathematical represen
From playlist Relativity