User:Jkasd/History of knot theory

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Knots have been used since antiquity, and people have known of the various properties of certain knots for centuries. Knots were first studied from a mathematical point of view by Carl Friedrich Gauss and his student Johann Benedict Listing.

Sir William Thomson (Lord Kelvin) theorized that atoms were knots of swirling vortices in the æther. This inspired Peter Guthrie Tait and others to try to classify all possible knots believing this would be equivalent to a periodic table of the elements. After Kelvin's vortex theory became obsolete, knot theory was no longer of great scientific interest.

After topology was founded, knots were investigated from a topological point of view, and some important discoveries were made such as the Alexander polynomial and the Reidemeister moves. Knot theory received a resurgence in mathematical interest after discoveries like the Jones polynomial and William Thurston's hyperbolization theorem.

Recently, new applications for knot theory have been discovered. Knots can be useful in detecting the chirality of molecules, and in studying the effects of topoisomerase on DNA. The related theory of braids, is the mathematical basis of topological quantum computers.

Beginnings[edit]

For thousands of years, knots have been used by sailors, climbers, and other people. Over time people realized that different knots were better at different tasks. Knots were studied by Carl Friedrich Gauss, who developed the Gauss linking integral for computing the linking number of two knots. His student Johann Benedict Listing, after whom Listing's knot is named, furthered their study.

Vortex theory[edit]

Trivial Knots

In 1867, Sir William Thomson (Lord Kelvin) came to the idea that atoms were knots of swirling vortices in the ætherafter observing Scottish physicist Peter Tait's experiments involving smoke rings. He thought that chemical elements might correspond to knots and links. Tait's experiments were inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids. Thomson and Tait believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do. For example, Thomson thought that sodium could be the Hopf link due to its two lines of spectra. (Modern physics demonstrates that the discrete wavelengths depend on quantum energy levels.) (Sossinsky 2002, p. 3-10)

Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. (The conjectures were finally resolved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and T. P. Kirkman. (Sossinsky 2002, p. 6)

James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also developed a strong interest in knots. Maxwell studied Listing's work on knots. He re-interpreted Gauss' linking integral in terms of electromagnetic theory. In his formulation, the integral represented the work done by a charged particle moving along one component of the link under the influence of the magnetic field generated by an electric current along the other component. Maxwell also continued the study of smoke rings by considering three interacting rings.

When the luminiferous æther was not detected in the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest.

Topological progress[edit]

Following the development of topology in the early 20th century spearheaded by Henri Poincaré, topologists such as Max Dehn, J. W. Alexander, and Kurt Reidemeister, investigated knots. Out of this sprang the Reidemeister moves and the Alexander polynomial.(Sossinsky 2002, p. 15-45) Dehn also developed Dehn surgery, which related knots to the general theory of 3-manifolds, and formulated the Dehn problems in group theory, such as the word problem. Early pioneers in the first half of the 20th century include Ralph Fox, who popularized the subject. In this early period, knot theory primarily consisted of study into the knot group and homological invariants of the knot complement.

Modern resurgence[edit]

A few major discoveries in the late 20th century greatly revived knot theory. In the late 1970s William Thurston's hyperbolization theorem introduced the theory of hyperbolic 3-manifolds into knot theory and made it of prime importance. In 1982, Thurston received a Fields Medal, the highest honor in mathematics, largely due to this breakthrough. Thurston's work also led, after much expansion by others, to the effective use of tools from representation theory and algebraic geometry. Important results followed, including the Gordon-Luecke theorem, which showed that knots were determined (up to mirror-reflection) by their complements, and the Smith conjecture.

Interest in knot theory from the general mathematical community grew significantly after Vaughan Jones' discovery of the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial, HOMFLY polynomial, and Kauffman polynomial.[1] Jones was awarded the Fields medal in 1990 for this work.(Sossinsky 2002, p. 71-89) In 1988 Edward Witten proposed a new framework for the Jones polynomial, utilizing existing ideas from mathematical physics, such as Feynman path integrals, and introducing new notions such as topological quantum field theory (Witten 1989). Witten also received the Fields medal, in 1990, partly for this work. Witten's description of the Jones polynomial implied related invariants for 3-manifolds. Simultaneous, but different, approaches by other mathematicians resulted in the Witten-Reshetikhin-Turaev invariants and various so-called "quantum invariants", which appear to be the mathematically rigorous version of Witten's invariants (Turaev 1994).

In the early 1990s, knot invariants which encompass the Jones polynomial and its generalizations, called the finite type invariants, were discovered by Vassiliev and Goussarov. These invariants, initially described using "classical" topological means, were shown by 1994 Fields Medalist Maxim Kontsevich to result from integration, using the Kontsevich integral, of certain algebraic structures (Kontsevich 1993, Bar-Natan 1995).

These breakthroughs were followed by the discovery of Khovanov homology and knot Floer homology, which greatly generalize the Jones and Alexander polynomials. These homology theories have contributed to further mainstreaming of knot theory. [2]

New applications[edit]

In the last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes called topoisomerases on DNA. [1]

In physics it has been shown that certain quasiparticles such as anyons and plektons, exhibit certain topological properties. Because the quantum states are left unchanged by ambient isotopy, hopes of making a quantum computer resistant to decoherence. Braid theory, a subset of knot theory, is used in studying the properties of such a computer, called a topological quantum computer. [3]

References[edit]

  1. ^ a b Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, 2001, ISBN 0-7167-4219-5
  2. ^ http://arxiv.org/pdf/math.QA/0201043.pdf
  3. ^ http://info.phys.unm.edu/~thedude/topo/sciamTQC.pdf

See also[edit]