Editing Knot theory (section)

==History==
{{main|History of knot theory}}
[[File:KellsFol034rXRhoDet3.jpeg|thumb|upright=.85|Intricate Celtic knotwork in the 1200-year-old [[Book of Kells]]]]
Archaeologists have discovered that knot tying dates back to prehistoric times.  Besides their uses such as [[khipu|recording information]] and [[knot tying|tying]] objects together, knots have interested humans for their aesthetics and spiritual symbolism.  Knots appear in various forms of Chinese artwork dating from several centuries BC (see [[Chinese knotting]]). The [[endless knot]] appears in [[Tibetan Buddhism]], while the [[Borromean rings]] have made repeated appearances in different cultures, often representing strength in unity.  The [[Celtic Christianity|Celtic]] monks who created the [[Book of Kells]] lavished entire pages with intricate [[Celtic knot]]work.

[[File:Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg|thumb|left|upright|The first knot tabulator, [[Peter Guthrie Tait]]]]
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. Mathematical studies of knots began in the 19th century with [[Carl Friedrich Gauss]], who defined the [[linking integral]] {{Harv|Silver|2006}}.  In the 1860s, [[William Thomson, 1st Baron Kelvin|Lord Kelvin]]'s [[Vortex theory of the atom|theory that atoms were knots in the aether]] led to [[Peter Guthrie Tait]]'s creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the [[Tait conjectures]].  This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of [[topology]].

These topologists in the early part of the 20th century—[[Max Dehn]], [[James Waddell Alexander II|J. W. Alexander]], and others—studied knots from the point of view of the [[knot group]] and invariants from [[Homology (mathematics)|homology]] theory such as the [[Alexander polynomial]].  This would be the main approach to knot theory until a series of breakthroughs transformed the subject.

In the late 1970s, [[William Thurston]] introduced [[hyperbolic geometry]] into the study of knots with the [[geometrization conjecture|hyperbolization theorem]]. Many knots were shown to be [[hyperbolic knot]]s, enabling the use of geometry in defining new, powerful [[knot invariant]]s. The discovery of the [[Jones polynomial]] by [[Vaughan Jones]] in 1984 {{Harv|Sossinsky|2002|pp=71–89}}, and subsequent contributions from [[Edward Witten]], [[Maxim Kontsevich]], and others, revealed deep connections between knot theory and mathematical methods in [[statistical mechanics]] and [[quantum field theory]]. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as [[quantum group]]s and [[Floer homology]].

In the last several decades of the 20th century, scientists became interested in studying [[physical knot theory|physical knots]] in order to understand knotting phenomena in [[DNA]] and other polymers. Knot theory can be used to determine if a molecule is [[chirality (chemistry)|chiral]] (has a "handedness") or not {{Harv|Simon|1986}}. [[tangle (mathematics)|Tangle]]s, strings with both ends fixed in place, have been effectively used in studying the action of [[topoisomerase]] on DNA {{Harv|Flapan|2000}}.  Knot theory may be crucial in the construction of quantum computers, through the model of [[topological quantum computation]] {{Harv|Collins|2006}}.