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Knot diagrammatics. (English) Zbl 1098.57005

Menasco, William (ed.) et al., Handbook of knot theory. Amsterdam: Elsevier (ISBN 0-444-51452-X/hbk). 233-318 (2005).
This paper is a survey of knot theory and invariants of knots and links from the point of view of categories of diagrams. The paper is divided into seven sections. In “Reidemeister moves” the author gives a proof of Reidemeister’s basic theorem (that the three Reidemeister moves on diagrams generate ambient isotopy of links in three-space). A discussion on graph embeddings extends Reidemeister’s theorem to graphs and proves the appropriate moves for topological and rigid vertices.
The third section – Vassiliev invariants and invariants of rigid vertex graphs – is expository, with discussions of the four-term relations, Lie algebra weights, relationships with the Witten functional integral and combinatorial constructions for some Vassiliev invariants. The section on the functional integral introduces the abstract tensor notation that helps understanding how the Lie algebra weight systems are related to the functional integral.
Section four and five are based on a reformulation of the Reidemeister moves so that they work with diagrams arranged generically transverse to a special direction in the plane. It is interesting how the technique by which the author proved Reidemeister’s theorem generalizes to give these moves as well. The moves with respect to a vertical are intimately related to quantum link invariants and Hopf algebras. Section four is a quick expositon of quantum link invariants, their relationship with Vassiliev invariants, the classical Yang-Baxter equation and infinitesimal braiding relations.
Section six is a discussion of the Temperley-Lieb algebra. The author gives a neat proof of the relation structure in the Temperley-Lieb monoid via piecewise linear diagrams. The last part of this section explains the relationship of the Temperley-Lieb monoid to parenthesis structures and shows how this point of view can be used to relate parenthesis to the pentagon and the Stasheff polyhedron. Section seven discusses virtual Knot theory, biquandles, Gauss diagrams and Vassiliev invariants. In the final section the author discusses the construction of links that while linked, have the same Jones polynomial as the unlink.
This is an interesting paper with topics that range from foundations of knot theory to virtual knot theory and topological quantum field theory and it also has an extensive bibliography.
For the entire collection see [Zbl 1073.57001].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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