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In mathematics , low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions . Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory , and braid groups . It can be regarded as a part of geometric topology . A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by Smale , in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory . Thurston's geometrization conjecture , formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics.
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type , developed 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 . A particular Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime.
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 with integer coefficients. [ edit ] Definition by the bracket Type I Reidemeister move Suppose we have an oriented link , given as a knot diagram .
Since its invention, Rubik’s Cube has taunted mathematicians trying to figure the maximum number of moves necessary to solve it from any of its 43,252,003,274,489,856,000 possible starting positions. Someone dubbed the effort a search for “God’s number,” ignoring the theological consensus that Einstein’s maxim “God does not play dice” is likely to apply to yo-yos, Slinkies, Rubik’s Cubes, and the whole range of handheld human amusements. Whatever you call it, the search has ended. In 2010 a team of whizzes laid bare the uplifting truth: As hopelessly scrambled as one’s cube may appear, one is never more than 20 moves from rendering each of its six faces a solid color. “We were secretly hoping in our tests that there would be one that required 21,” team member Morley Davidson, a mathematician at Kent State University, told the BBC.
An ideal chain (or freely-jointed chain ) is the simplest model to describe a polymer . It only assumes a polymer as a random walk and neglects any kind of interactions among monomers . Although it is simple, its generality gives us some insights about the physics of polymers. In this model, monomers are rigid rods of a fixed length l , and their orientation is completely independent of the orientations and positions of neighbouring monomers, to the extent that two monomers can co-exist at the same place. [ edit ] The model N monomers form the polymer, whose total unfolded length is: