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Wirtinger presentation

From Wikipedia, the free encyclopedia

In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form where is a word in the generators, Wilhelm Wirtinger observed that the complements of knots in 3-space have fundamental groups with presentations of this form.

Preliminaries and definition

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A knot K is an embedding of the one-sphere S1 in three-dimensional space R3. (Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot, is the knot complement. Its fundamental group is an invariant of the knot in the sense that equivalent knots have isomorphic knot groups. It is therefore interesting to understand this group in an accessible way.

A Wirtinger presentation is derived from a regular projection of an oriented knot. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing.

Wirtinger presentations of high-dimensional knots

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More generally, co-dimension two knots in spheres are known to have Wirtinger presentations. Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied:

  1. The abelianization of the group is the integers.
  2. The 2nd homology of the group is trivial.
  3. The group is finitely presented.
  4. The group is the normal closure of a single generator.

Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem.

Examples

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For the trefoil knot, a Wirtinger presentation can be shown to be

See also

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Further reading

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  • Rolfsen, Dale (1990), Knots and links, Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish, ISBN 978-0-914098-16-4, section 3D
  • Kawauchi, Akio (1996), A survey of knot theory, Birkhäuser, doi:10.1007/978-3-0348-9227-8, ISBN 978-3-0348-9953-6
  • Hillman, Jonathan (2012), Algebraic invariants of links, Series on Knots and Everything, vol. 52, World Scientific, doi:10.1142/9789814407397, ISBN 9789814407397
  • Livingston, Charles (1993), Knot Theory, The Mathematical Association of America