Saturday, March 12, 2011

Knot Theory (and the Jones Polynomial)



It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS 3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

... The Abstract to Ed Witten's  "Quantum Field Theory and the Jones Polynomial". Commun. Math. Phys. 121 (3): 351–399. MR0990772. 1989

from various Wiki-places:

The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type, introduced by Edward Witten. It is so named because its action is proportional to the integral of the Chern–Simons 3-form.

In condensed matter physics, Chern–Simons theory describes the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.

(Edward Witten argued that the Kodama state in loop quantum gravity is unphysical due to an analogy to Chern–Simons state resulting in negative helicity and energy. There are disagreements to Witten's conclusions.)
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In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.[1] Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable t1 / 2 with integer coefficients.

Steve here. Well, that's the Wiki definition. I like the way Clifford A. Pickover in  The Math Book puts it better:

      In Mathematics, even the most tangled loop in three dimensions can be representented as a projection, or shadow, on a flat surface. When mathematical knots are diagrammed, tiny breaks in the lines often indicate when a strand crosses over or under another strand. 

     One of the goals of knot theory is to find invariants of knots, where the term invariant refers to a mathematical characteristic or value that is the same for equivalent knots so that it can be used to show that two knots are different. In 1984, knot theorists were all abuzz with a startling invention of New Zealand mathematician Vaughn Jones, an invariant, now called the Jones Polynomial, that could distinguish more knots than any previous invariant. Jones had made his breakthrough discovery by chance, while working on a Physics problem. Mathematician Kevin Devlin writes, "Sensing that he had stumbled onto an unexpected, hidden connection, Jones consulted knot theorist Joan Birman, and the rest, as they say, is history ..." Jones research "opened the way to a whole array of new polynomial invariants, and led to a dramatic rise in research in knot theory, some of it spurred on by the growing awareness of exciting new applications in both biology and physics ..." Biologists who study DNA strands are interested in knots and how they can help elucidate the functioning of genetic material in cells or even aid in resistance to viral attacks. A systematic procedure, or algorithm, allows mathematicians to express the Jones polynomial for any knot, based on its patterns of crossings.

      The use of knot invariants has had a long history. Around 1928, James W. Alexander (1881-1971) introduced the first polynomial associated with knots. Alas, the Alexander polynomial was not usable for detecting the difference between a knot and its mirror image, something that the Jones polynomial could do. Four months after Jones announced his new polynomial, the more general HOMFLY polynomial was announced.

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In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of Sj in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory.

A table of all prime knots with seven crossings or fewer (not including mirror images).
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In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); 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. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

2 comments:

Jérôme CHAUVET said...

Dear Steven,

Fascinating topic indeed. I guess I could spend hours trying to untangle complicated knots, just for the fun of it.

That's interesting to see how the human brain has it hard to figure out whether or not two apparently different knots are in fact the same knot for which the rope has undergone a sufficient displacement between the two instances.

The fact that the brain has it hard to perform this mental untanglement proves this is an important theory, as intuition is in these problems definitely unhelpful.

Best,

Steven Colyer said...

I hear you, but intuition IMO is a guidepost, like an Indian guide in America's Old West, that shows the way. Lewis and Clark still have to get on their horse and go there, which is my analogy for actually doing the mathematical equations.

And the human brain IMO is one big knot. A wonderfully complicated knot, but still a knot.

Still much work to be done. QUITE a bit. Computers should help.

Personally, I didn't know that quantum Hall effect uses this stuff. Since qHe is my default specialty (in Physics) should my flaky brain continue to have trouble PICKING a specialization, I think I shall study this more.