Theory of the nodes
See also: Node
The Théorie of the Nœud S is a branch of the Topologie which consists of the mathematical study of idealized pieces of string. The theory of the nodes is thus very close to the Théorie of the braids.
History
The theory of the nodes has a long story. One can perhaps make it start with work of Gauss related to the electromagnetism.
The theory of the nodes is always under development full.
Tally mathematical
More formally, one considers plungings of the Cercle in the Euclidean Espace of dimension 3. A node is such a plunging considered except for deformation (or Isotopie). They are thus rather here strings without ends that nodes with the usual direction.
Equivalence of nodes
The principal problem is to determine if two different plungings are in fact the same node. For that, it is advisable to build invariants of the nodes, which are functions on the whole of plungings which depend only on the node. Once defined an invariant, it will still be necessary to seek to know up to what point it takes different values on different nodes.
Invariants of nodes
Among the principal invariants of the nodes, let us quote the Polynôme of Jones, the Polynôme of Alexander, the Polynôme HOMFLY, the fundamental Groupe of complementary to the node, the invariants of the type finished of Vassiliev and the integral of Kontsevich.
Among the last introduced invariants, there are in particular groups of homology of nodes defined by Khovanov.
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