The Tangle Category
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The aim of this chapter is to set up a categorical construction of isotopy invariants of links. To this end, we build a strict tensor category T out of the tangles defined in X.5. Any strict tensor functor from T to a category of finite-dimensional vector spaces gives rise to an isotopy invariant. Using a presentation of T by generators and relations, we shall reduce in Section 4 the construction of such a functor to an algebraic data, called an enhanced R-matrix, consisting of a finite-dimensional vector space, an R-matrix, and a compatible automorphism. We shall apply this method in Section 5 to exhibit explicit isotopy invariants that will allow us to complete the proof of Theorem X.4.2 asserting the existence of the Jones-Conway polynomial.
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