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In this coffee break, I present you a brief historical account of knot theory so that you will be relaxed and intriguied to dig more out of the core of the theory.
It was Gauss (why always Gauss?) who initiated the study of knots and links. He created some basic calculus used to distinguish some knots and links from one another. More amazingly, he found a relation between the linking number of links and the electrogmagetism (we will come back to this point later). He considered knot theory as analysis situs and even predicted that it will become a main branch of future mathematics but not of the mathematics of his time. After this, the study of knot theory had been silent for a little while until Lord Kelvin proposed in 1875 that atoms could be knots of votex lines of ether. Because of this, studying knots was a kind of hip at that period of time. A Physicist, P. G. Tait was first inspired by Kelvin's theory to study knots in detail. He finished tabulating knots, according to the number of crossings (up to 8 crossings) in 1885a. This is actually a great work. Although such an atomic theory died completely when Lord Kelvin admitted his failure, the mathematicl development of knot theory survived. However, in the origianl Tait's knot table there are some knots that are actually the same but just can not be recogonized by known mathematical tools at his time. Tait then published the famous Tait's conjecturesb for classifying the knots/links by manipulating their plane projections.
Mathematicians liked conjectures. They then started to move their hands on knots/links. In 1926, K. Reidemeister proved a wonderful theorem saying that two knots are isotopic to each other if and only if one can be deformed into the other by three simple local moves, now christened Reidemeister moves. We will get back to this soon in the next post. However, this is still a graphical method, which turns to be inconvenient on manipulating complicated knots. Well, fine, mathematicians never stop. They came to think about algebraic objects associated with knots/links that are invariant under isotopyc. J. W. Alexander came up with the first good such quantity, the Alexander polynomiald. However, people soon realized that this polynomial was not powerful enough in that many non-isotopic knots share the same polynomial and that it can not distinguish links from their mirror images. Things became more interesting. There turned out to be two apparently different routes, both of which lead to astonishing findings of the connections between knot theory and physics.
Along the first track, people had still been looking for polynomials. Tens of year passed. The first exicting moment rang in the mid of 1980's. In his 1985 papere, V. Jones openned up a brand new era of knot theory by introducing to the society a new and paramount polynomial, the Jones polynomial as what we know now. This discovery was somehow an accident because Jones was actually working on operator theory. Nevertheless, it was exactly the operator teory that helped him to obtain this distinct polynomial from Alexander-Conway polynomial. A topologist, J. Birman in one of Jones's seminar pointed out that some equations Jones wrote on the blackboard looked alike some formulae in knot theory. Jones were interested and started his research on this similarity and later he found out the polynomial with the help of Von Neumann algebra. If you know about quantum mechanics and/or quantum field theory, you may notice the close relation between them and the operator theory. Now the threads weaving knot theory and physics together seems clearer and clearer. We are getting there soon. But wait for a second, allow me a bit more verbose. With Jones polynomial, people soon found out the proof of Tait's conjectures, which had bothered mathematicians about 100 years. Jones polynomial has many remarkable properties, for example its great capability in distinguishing some knots from their mirror imagesf. These properties, which were not comprehensible within the scope of existing algebraic topological theory, stimulated a mass endeaver of people in searching and understanding new polynomials. Shortly after, a couple of polynomials. e.g. two variable Jones polynomial and HOMFLYg (also known as LYMPHTOFU) polynomial, were born as generalizations of Jones polynomial. By the way, Jones's work brought many mathematicians and physicists jobs, and hence he won the Fields medal of 1990. Although Jones polynomial was not complicated, it still involved advanced mathematics. In 1987, L. H. Kauffmanh gave an equivalent form of Jones polynomial in terms of elementary math and thus made it understandable to almost anybody. Kauffman adopted a sort of state model in analogy to the partition function of a 2-dimensional statistical system such as an Ising model. Behind this analogy, is there indeed a deep mutual-beneficial relation between knot theory and 2-dimensional statistical mechanicsi.
Ok, Mathematicians had too much fun already. It is now physicists' turn. History seems like to repeat similar things. This time again, physicists showed to the world a fascinating and revolutionary picture by taking the adavantage of their peculiar physical insights and what mathematician had done on knot theory and on other mathematical branches. E. Witten, a famous physicist, played a most important role in this revolution with no doubt. Recall that Gauss found the relation between the linking number, a simplest invariant, and electromagnetism, which is now known as a 
gauge theory. The fact that this is absolutely not a conincidence was unraveled by Witten who showed that each link invariant corresponds to each finite-dimensional representation of each semisimple Lie group. While the linking number is associated with abelian gauge group, the group, other link invariants can find their partners among non-abelian gauge groups. Our major role in previous paragraph, the Jones polynomial is the very simplest polynomial invariants due to its correspondance to the lowest representation—the spin-1/2 representation— of the group 
. What a fantabulous discovery this is! Quantum field theories describing the four fundamental interactions, viz electromagnetic, weak, strong, and gravitational interactions, are essentially gauge field theoriesj. That is to say, Witten's work related knot theory, which belongs to low-dimensional topological theories, to quantum field theories that describe the very physical side of the world. This new angle of viewing quantum field theories inspired a burst of scientific activities in this direction. As usual, this is not yet the end of the story. However, Iet us time travel back to tens of years ago and run on the other route of discoveries. Amazingly, we will re-arrive at the same point where we get from the first route and we will see a more exciting picture..
To be continuted…
Footnotes & References:
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He tried to plot in his table the knots (projections of paper, of course) that are prime knots non-isotopic to each other. The definition of prime knots will be given later in the next post in this serie. But due to the lack of mathematical tools, in his original table, there were knots which are actually isotopic to each other. He also ignored the unknot, which is the trivial circle with 0 crossing. It is obvious that a non-trivial knot must have number of crossings greater than or equal to 3. There are other knot tables made by people in the 20th century, for example the famous Rolfsen table by D. Rolfsen. follow this link: http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table , you can see pictures of the Rolfsen table and the Hoste-Thistlethwaite Table.
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These conjecture had bot been proven until late 1980's due to the discovery of Jones polynomial.
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These are the so called knot/link invariants, a type of topological invariants which are usually numbers or polynomials of single or multiple variables. A topological invariant is a quantity associated with a topology. All the topologically same spaces, i.e. homeomorphic spaces share the same value of that quantity. simple logic then tells us that if two topological spaces have different value of the same topological invariant they must be inhomeomorphic to each other. However, two spaces with the same value of a topological invariant are not necessarily homeomorphic. It might be better for you to understand If you know Euler characteristic, a simple but important topological invariant of topological spaces.
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It is now called Alexander-Conway polynomial because its proper normalization and simpler definition was given by Conway in 1970.
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V. Jones, "A polynomial invariant of knots and links", Bull. Amer. Math. Soc., 12 , pp103-111, 1985.
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Jones polynomial is the first polynomial which can do this.
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HOMFLY is the acronym of the last names of six different mathematicians due to their simultaneous but independent discovery of the polynomial. One may see a bit from this that there was really a big mass activity inspired by Jones's work.
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L. Kauffman, "state models and Jones polynomial", Topology, 26, pp395-407, 1987.
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There are many other connections of knot polynomials to two (1+1) –dimensional physics, a summary of these relations can be found in in Witten's 1989 paper: [Commun. Math. Phys. 121, 351–399 (1989)] and references therein.
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Quantum field theories of the other three fundamental interactions than gravity have long been recognized as gauge field theories and have been unified in a sense within the framework of the so-called standard model. People have not realized that the theory of gravity can also be described as a gauge theory until A. Ashtekar in 1986 discovered in his great paper that this is indeed the case by reformulating general relativity in terms of his new set of variables, now known as Ashtekar variable. However, there is still some subtlety, which is out of the scope of this aritle and will actually be discussed in another post in this blog. Consequently, some people started to quantize the gravity by utilizing this new formalism and produced the Loop Quantum Gravity, one the two main streams of quantum theory of gravity.
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