History of knot theory
In 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther.
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.
[1]: 6 James Clerk Maxwell, a colleague and friend of Thomson's and Tait's, also developed a strong interest in knots.
Following the development topology in the early 20th century spearheaded by Henri Poincaré, topologists such as Max Dehn, J. W. Alexander, and Kurt Reidemeister, investigated knots.
In the early 1970s, Friedhelm Waldhausen announced the completion of Haken's program based on his results and those of Klaus Johannson, William Jaco, Peter Shalen, and Geoffrey Hemion.
A few major discoveries in the late 20th century greatly rejuvenated knot theory and brought it further into the mainstream.
In 1982, Thurston received a Fields Medal, the highest honor in mathematics, largely due to this breakthrough.
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.
[1]: 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.
In the last several decades of the 20th century, scientists and mathematicians began finding applications of knot theory to problems in biology and chemistry.
More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula.
In physics it has been shown that certain hypothetical quasiparticles such as nonabelian anyons exhibit useful topological properties, namely that their quantum states are left unchanged by ambient isotopy of their world lines.
