Knot theory

Knot theory is a research field of topology. It deals, among other things, to investigate the topological properties of nodes. One question is about whether two given nodes are equivalent, ie whether they can be transformed into one another, without changing the string " cut " is. Knot theory is not concerned as opposed to knots with the knot tying in practice, but with mathematical structures.

Mathematical definition

In the mathematical sense, a node is an embedding a circular line in the three-dimensional space. Two nodes are considered equal if they can be converted into each other by a continuous deformation ( isotopy ).

In knot theory and embeddings of multiple circuit lines are investigated; these are called tangles. Another extension of the theme are multidimensional node, that is embeddings of spheres of dimension n in the n 2 dimensional space for n> 1

Knot diagrams and Reidemeister moves

In knot theory, a knot is often represented by its projection onto a plane. The projection is chosen so that it has only a finite number of double points ( intersections ). Nodes at which you can not specify such a projection is called a wild node ones. Having such a projection as tame nodes For example, each piecewise smooth knot is tame. In knot theory mainly properties are investigated of tame knots.

In order to reconstruct a projection node, you must specify at each intersection, which is the two strands up and down. A projection of this additional information is called a knot diagram. Each node tame thus can be represented by a diagram. However, such a graph is not unique, because each node can be represented by an infinite number of different diagrams. For example, the following local trains change, although the diagram, but not the nodes shown:

These trains are called Reidemeister moves, in honor of Kurt Reidemeister. This has shown in 1927 that these three traits suffice: Two node graphs show exactly the same node then is if they can be converted by a finite sequence of Reidemeister moves together.

Knot invariants

One goal of knot theory is to find node - invariants, ie mathematical objects that do not change if one deforms the nodes in three-dimensional space steadily. Some examples of knot invariants are polynomials, such as the Jones polynomial HOMFLY polynomial or Kauffman polynomial. These polynomials can be calculated algorithmically using a graph of nodes by adding suitable for all intersections Terme to a Gesamtpolynom. For invariance it suffices to show that the polynomial constructed in this way is invariant under the three Reidemeister moves.

To date, no simple predictable node - invariant been found that distinguishes all non-equivalent nodes, ie which has the property that the invariant for two nodes is identical if and only if the two nodes are equivalent. This is an important goal of current research (solutions for this, there may be in the area of ​​Floer homology ). It is also unknown whether the Jones polynomial detects the trivial knot, ie whether there is a non -trivial node whose Jones polynomial is equal to the trivial knot.

Applications

For a long time employment with nodes rather an unprofitable art. But in the meantime there are a number of important applications, such as biochemistry and structural biology, which can be checked whether complicated folds of proteins are consistent with other proteins. The same applies to the DNA. Other current applications are available in polymer physics. Great importance is given knot theory in modern theoretical physics, where it is about to paths in Feynman diagrams.

Knot theory is also used in neighboring areas of topology and geometry. To study three -dimensional spaces are very useful node, since each closed orientable 3 -dimensional manifold can by Dehn surgery to create a node or entanglement. In hyperbolic geometry nodes play a role, because carry a complete hyperbolic metric of the complements of the most nodes in the 3-dimensional sphere.