Braided compact closed categories with applications to low dimensional topology (Q1822802)

The discovery by \textit{V. F. R. Jones} [Bull. Am. Math. Soc., New. Ser. 12, 103-111 (1985; Zbl 0564.57006)] of a new isotopy invariant of classical links, and its subsequent generalizations [\textit{P. Freyd}, \textit{D. N. Yetter}, \textit{J. Hoste}, \textit{W. B. R. Lickorish}, \textit{K. Millet} and \textit{A. Ocneanu}, Bull. Am. Math. Soc., New Ser. 12, 239-246 (1985; Zbl 0572.57002); \textit{L. H. Kauffman}, Am. Math. Mon. 95, No.3, 195-242 (1988; Zbl 0657.57001) and ``On knots'' (1987; Zbl 0627.57002)], has lead to renewed interest in algebraic representations of knotting and linking. Prompted by this, \textit{D. N. Yetter} [Contemp. Math. 78, 705-730 (1988; Zbl 0665.57004)] analyzed the structure of ``categories of tangles'' to give a combinatorial description of composition of generalized tangles. In that work a great simplification was achieved by using a monoidal structure on the categories considered. As we shall see, the assumption in [Yetter, loc. cit.] that links (or more generally ``Markov classes'') arise by the imposition of extrinsic structure (i.e. Markov moves) is unnecessary when the full categorical structure (including the correct generalization of compact closedness to the non-symmetric case) is understood, an observation first made by \textit{P. Freyd}. This observation leads to a clear representation-theoretic view of the new link invariants and, when coupled with the encoding of framings by a modified diagrammatic ``isotopy'' rather than the labels used in [Yetter, loc.cit], to an ``algebraicized'' Kirby calculus from which it is actually possible to calculate invariants of 3-manifolds from the surgery presentation.