Low-dimensional topology. 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.
Chern–Simons theory. 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. The classical theory[edit] Mathematical origin[edit] In the 1940s S.
S. Chern and A. Weil studied the global curvature properties of smooth manifolds M as de Rham cohomology (Chern–Weil theory), which is an important step in the theory of characteristic classes in differential geometry. . In 1974 S. Where T is the Chern–Weil homomorphism. Where C is a (2k − 1)-dimensional cycle on M. Where is the first Pontryagin number and s(M) is the section of the normal orthogonal bundle P. The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. Configurations[edit] Chern–Simons theories can be defined on any topological 3-manifold M, with or without boundary.
Dynamics[edit] by. Jones polynomial. With integer coefficients.[2] Definition by the bracket[edit] Type I Reidemeister move , given as a knot diagram. We will define the Jones polynomial, , using Kauffman's bracket polynomial, which we denote by . Note that here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) where denotes the writhe of in its given diagram. In the figure below) minus the number of negative crossings ( ).
Is a knot invariant since it is invariant under changes of the diagram of by the three Reidemeister moves. Under a type I Reidemeister move. Polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves. Now make the substitution in to get the Jones polynomial . Definition by braid representation[edit] Jones' original formulation of his polynomial came from his study of operator algebras.
Let a link L be given. And. . , where. #95: Rubik’s Cube Decoded | Math. Ideal chain. An ideal chain (or freely-jointed chain) is the simplest model to describe polymers, such as nucleic acids and proteins. 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. The model[edit] N monomers form the polymer, whose total unfolded length is: , where N is the number of monomers. In this very simple approach where no interactions between monomers are considered, the energy of the polymer is taken to be independent of its shape, which means that at thermodynamic equilibrium, all of its shape configurations are equally likely to occur as the polymer fluctuates in time, according to the Maxwell–Boltzmann distribution.
Since and.