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Topological helicity for framed links. (English) Zbl 1064.57011

In this article an invariant called topological helicity is defined for oriented framed links. The framing can be realized in a diagram by drawing each component of the link as a ribbon whose boundaries inherit the orientation of the link. If we assign to each part of a ribbon crossing another part of a ribbon a number \(\pm 1\) depending on orientation and to each half twist in a ribbon a number \(\pm 1/2\) depending on the direction of twisting then the topological helicity can now be computed by summing the various signed numbers associated to the crossings in the diagram. This sum will be an invariant of oriented framed links because it can be interpreted as a sum of linking numbers of the ribbon boundaries. Topological helicity provides a concise, purely topological description of helicity of magnetic flux ropes. Helicity is a well-known invariant for linked magnetic flux ropes and usually computed by means of an integral. Two magnetic flux ropes may undergo a process of magnetic reconnection by which magnetic field lines in close proximity are broken apart and reconnected. This process can be modeled on the set of oriented framed links by introducing an equivalence relation called reconnection. It is shown that two oriented framed links are reconnection-equivalent if and only if they have the same topological helicity.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
82D10 Statistical mechanics of plasmas
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