Knot theory is the branch of topology that studies mathematical knots, embeddings of a circle in three-dimensional space whose ends are joined so they cannot be undone. It emerged from nineteenth century physical speculation that atoms were knotted vortices in the ether, which motivated the first systematic knot tables in the 1880s, and it survived the collapse of that physical theory to become a mathematical field in its own right.
Facts
Central QuestionGiven two different-looking descriptions of a closed loop embedded in three-dimensional space, is there a way, ideally a computable one, to tell whether they represent the same knot? 1 Key DebateWhether knot theory's founding motivation, Lord Kelvin's nineteenth century theory that atoms were stable knotted vortices in the ether, deserves credit as a genuine scientific origin given that the physical theory itself was entirely wrong; the systematic knot tables Peter Guthrie Tait built to catalogue possible vortex atoms in the 1880s outlived the physics that inspired them and became the field's own foundation. 1 Classification
Pure or Applied Connections
Associated With
Source Peter Guthrie Tait, Biography (MacTutor History of Mathematics)
Includes
Source Alexandre-Théophile Vandermonde (Wikipedia)
Source Wolfram MathWorld: Borromean Rings
Source Wikipedia: Braid group
Source Cinquefoil Knot (Wikipedia)
Source Conway knot (Wikipedia)
Source Wikipedia: Figure-eight knot (mathematics)
Source Herbert Seifert, Biography (MacTutor History of Mathematics)
Source Hopf Link (Wikipedia)
Source Wolfram MathWorld: Knot
Source Maxim Kontsevich (Wikipedia)
Source Reidemeister move (Wikipedia)
Source Ruth Lawrence (Wikipedia)
Source Wolfram MathWorld: Seifert Surface
Source Wikipedia: Trefoil knot
Source Volume Conjecture (Wikipedia)
Source Whitehead link (Wikipedia)
Source Wilhelm Wirtinger (Wikipedia)
Source Writhe (Wikipedia)
Sources
1. Knot Theory (Wikipedia)
WikipediaDefinition and history sections
a knot is an embedding of a circle in 3-dimensional Euclidean space
Central questions, the recognition problem
two descriptions represent the same knot
Historical origins section
Lord Kelvin's theory that atoms were knots in the aether
History section
A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position
In Group: Manifolds, Knots and Geometric Topology, lead paragraph
In topology, knot theory is the study of mathematical knots.
View the Source Wilhelm Wirtinger (Wikipedia)
Includes: Wilhelm Wirtinger, Lead, knot theoryQuote, Includes: Wilhelm Wirtinger, Lead, knot theory
working in complex analysis, geometry, algebra, number theory, Lie groups, and knot theory
Maxim Kontsevich (Wikipedia)
Includes: Maxim Kontsevich, Academic career and researchQuote, Includes: Maxim Kontsevich, Academic career and research
most notably on knot theory, quantization, and mirror symmetry
Ruth Lawrence (Wikipedia)
Includes: Ruth Lawrence, LeadQuote, Includes: Ruth Lawrence, Lead
a researcher in knot theory and algebraic topology
Alexandre-Théophile Vandermonde (Wikipedia)
Includes: Alexandre-Théophile Vandermonde, Article textQuote, Includes: Alexandre-Théophile Vandermonde, Article text
presaged the development of knot theory
Wikipedia: Trefoil knot
Includes: Trefoil Knot, Lead sentenceQuote, Includes: Trefoil Knot, Lead sentence
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.
Wikipedia: Figure-eight knot (mathematics)
Includes: Figure-Eight Knot, Lead sentenceQuote, Includes: Figure-Eight Knot, Lead sentence
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four.
Whitehead link (Wikipedia)
Includes: Whitehead Link, Lead sentenceQuote, Includes: Whitehead Link, Lead sentence
In knot theory, the Whitehead link, named for J.
Writhe (Wikipedia)
Hopf Link (Wikipedia)
Includes: Hopf Link, Lead sentenceQuote, Includes: Hopf Link, Lead sentence
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component.
Cinquefoil Knot (Wikipedia)
Includes: Cinquefoil Knot, Lead sentenceQuote, Includes: Cinquefoil Knot, Lead sentence
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing nu
Conway knot (Wikipedia)
Includes: Conway knot, Lead sentenceQuote, Includes: Conway knot, Lead sentence
In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named afte
Volume Conjecture (Wikipedia)
Reidemeister move (Wikipedia)
Includes: Reidemeister's Theorem, Lead sentenceQuote, Includes: Reidemeister's Theorem, Lead sentence
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram.
Wikipedia: Braid group
Includes: Braid Group, Lead paragraphQuote, Includes: Braid Group, Lead paragraph
Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang-Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.
Wolfram MathWorld: Seifert Surface
Wolfram MathWorldIncludes: Seifert Surface, Subject classificationsQuote, Includes: Seifert Surface, Subject classifications
Topology Knot Theory
Wolfram MathWorld: Knot
Wolfram MathWorldIncludes: Knot (Mathematics), Subject classificationsQuote, Includes: Knot (Mathematics), Subject classifications
Topology Knot Theory
Wolfram MathWorld: Borromean Rings
Wolfram MathWorldIncludes: Borromean Rings, Subject classificationsQuote, Includes: Borromean Rings, Subject classifications
Topology Knot Theory
Herbert Seifert, Biography (MacTutor History of Mathematics)
Includes: Herbert Seifert, SummaryQuote, Includes: Herbert Seifert, Summary
Herbert Seifert was a German mathematician who made important contributions to topology and knot theory.
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