What is Knot theory: Definition and 19 Discussions

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space,



{\displaystyle \mathbb {R} ^{3}}
(in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of



{\displaystyle \mathbb {R} ^{3}}
upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.
The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

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  1. math-physicist

    Possible research directions for a beginner in TQFT

    I am highly interested in Topological Quantum Field Theory (TQFT) and am currently planning on doing a project on this topic this year. Some of my relevant background: Algebra (Groups, Rings, Fields, basics of Categories and Modules), Topology (Munkres), Smooth Manifolds (John Lee's book, first...
  2. sophiatev

    A PD code yields two different knot diagrams

    The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, while a link diagram does not have a unique PD code, a given PD code should map to just one knot...
  3. E

    I Simpler Brunnian "rubberband" loops?

    The standard configuration of Brunnian "rubberband" loops shows a series of unknots each bent into a U-shape, with their ends looped around the middle of the next unknot. (See for instance http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links). This connection requires 8...
  4. B

    I Citation for a knot theory statement

    Hello! Where can I find the source for this statement (i.e. a citation for it, ideally the original one): "any smooth k-sphere embedded in ##R^n## with 2n − 3k − 3 > 0 is unknotted". Thank you!
  5. DaveC426913

    B Knot theory: closed loops in n dimensions

    I'm not a math machine, but I dabble in dimensional stuff. I think this falls under knot theory. I have built several prototypes of a tesseract. Each of them sits in a little case in my office. One of them is made from truncated cubes, held together with elastic cord: In theory, the...
  6. G

    Best Written High School Physics Text Books (SAT)

    Advanced Physics (Advanced Science) by Steve Adams & Jonathan Allday from OUP Oxford: and Physics (Collins Advanced Science) 3rd Edition by Kenneth Dobson from Collins Educational: http://www.amazon.com/dp/0007267495/?tag=pfamazon01-20 Does anyone know any of these books? I find them very...
  7. dkotschessaa

    Is "Kirby's Problem List" fairly familiar to Knot Theorists?

    I refer to the list here: https://math.berkeley.edu/~kirby/ entitled "Problems in Low-Dimensional Topology" Question says it all. I am giving a graduate level presentation to a group that includes some knot theorists. (I believe the collective term for knot theorists is a "tangle"...
  8. L

    Pictures of Magnetic Field Lines

    I have looked in vain on the web for pictures of magnetic field lines for multiple linked current loops. I would be happy just to see a picture of the field lines for a simple Hopf link but somewhere there must be pictures for the Borromean rings and other more complex links - and also braids...
  9. J

    What Are Some Current Topics in Physics That Interest John Salkeld?

    The title above give my name. I am a pure maths PhD with an interest in physics and geometry. I am currently studying physics for fun and I am very interested in current progress. I am especially interested in quantisation of space time, holographic theories and dualities. Regards John
  10. M

    Knot Theory: Exploring its Math & Non-Math Applications

    How is knot theory involved in other fields of math? Any applications outside of math?
  11. W

    Can anyone suggest an introductory book on knot theory?

    I want one good for physicists does not need to be very rigorous
  12. shreyakmath

    Applying of knot theory to string theory

    Probably a bit abstract,but I was thinking if 4D closed strings could form knots? I mean if a closed string in 4-dimensional spacetime can be considered an unknot and a knot polynomial be associated with every closed string. I also wondered that if the fundamental strings vibrate in the knotted...
  13. M

    Exploring 4D Knot Theory: Reidermeister Moves & Movie Representation

    I need a free description with illustrations on 4D knots theory, especially the 4D generalization of Reidermeister moves and the movie represantation. Where can I find a freely available paper?
  14. B

    Knot Theory Book: Intro for Topology & Algebraic Topology

    I'm looking for an introductory book for knot theory. I have background in topology and algebraic topology. I would prefer a more sophisticated treatment, but I have no previous knowledge.
  15. G

    Knot Theory and higher dimensions

    Hi, I was thinking about Knot Theory for a while and started thinking about higher dimensionalities. Could the knots we know so well (knots in 3d space) be undone if allowed to be manipulated through a fourth spatial dimension? Could they be made topologically equivalent to the unknot? And if...
  16. S

    Quandles are associated with knot theory

    So I know that quandles are associated with knot theory, etc. but what are "shadow colorings" ? on a broader context, can someone please give me a simple definition of a "knot invariant" and a "quandle" ? and what does this have to do with "racks" ? sorry, all this esoteric language is...
  17. D

    Is the Linking Number Always an Integer in Knot Theory?

    i cannot find a proof anywhere to show that the linking number is always an integer! can someone please point me in the right direction (or give a proof of their own)! thanks.
  18. D

    Can Two-Moves Separate Components of a Link in Knot Theory?

    Hi, so I need to show that every link is two-equivalent to a trivial link with the same number of components. Right now I can show that if I have a simple link with linking number = 1, then it is possible to immediately separate the link into its two components. But how can I generalize this...
  19. K

    Knot Theory & Parlor Games: Explained for High School Students

    Hi, I'll attend a lecture around 14 hours later addressed by a Nobel laureate and here's the brief description of the lecture. "Starting from a Parlor Game, I shall show how a deep mathematical problem can be formulated in an elementary way. The steps are understandable to high school...