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Virtual knot theory. (English) Zbl 0938.57006

Virtual knot theory is a generalization of classical knot theory; its objects are virtual knot diagrams which are 4-regular (4-valent) planar graphs, with an extra structure at the nodes: the usual under- and overcrossings plus a new type of crossing called virtual (the idea is that it is not really there). The equivalence of virtual knots and links is defined combinatorially by generalized Reidemeister moves involving also virtual crossings. The motivation for virtual knot theory comes from two sources. The first is the study of knots in thickened surfaces of higher genus (the classical case being that of the 2-sphere), the second the extension of knot theory to the purely combinatorial domain of Gauss codes and Gauss diagrams (to represent knots and links; nonplanar Gauss codes give rise to virtual knots).

In the present paper, the fundaments of virtual knot theory are developed, giving motivations and many examples. The fundamental group and its quandle generalization are discussed for virtual knots. Examples for various non-classical phenomena are given. There are non-trivial virtual knots with trivial group (i.e. the integers); some virtual knots are distinguished from their mirror images by the fundamental group. The bracket and Jones polynomials, quantum and Vassiliev invariants are discussed for virtual knots. There are non-trivial virtual knots with trivial Jones polynomial, infinitely many virtual knots with the same fundamental group, and a knotted virtual with trivial group and unit Jones polynomial. As noted at the end of the paper, the work began with an attempt to understand the Jones polynomial for classical knots by generalizing that category, in the hope that the considerations will lead to a deeper insight into the Jones polynomial and its relationship with the fundamental group and the quandle of a classical knot.

57M25Knots and links in the 3-sphere
57M27Invariants of knots and 3-manifolds