page_title: Knot theory
section_title: History
hierarchical_section_title: Knot theory / History
caption_attribution_description: Peter Guthrie Tait (April 28, 1831 - July 4, 1901).
context_page_description: 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. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself; 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.
context_section_description: Archaeologists have discovered that knot tying dates back to prehistoric times.  Besides their uses such as recording information and 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 monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.
  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 (Silver 2006).  In the 1860s, Lord Kelvin's 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, J. W. Alexander, and others—studied knots from the point of view of the knot group and invariants from 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 hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 (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 groups and Floer homology.
  In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not (Simon 1986). Tangles, strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA (Flapan 2000).  Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation (Collins 2006).
caption: The first knot tabulator, Peter Guthrie Tait
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