By Christian Kassel, Vladimir Turaev

During this well-written presentation, inspired via various examples and difficulties, the authors introduce the fundamental conception of braid teams, highlighting a number of definitions that convey their equivalence; this can be by means of a therapy of the connection among braids, knots and hyperlinks. very important effects then deal with the linearity and orderability of the topic. correct extra fabric is incorporated in 5 huge appendices. Braid teams will serve graduate scholars and a couple of mathematicians coming from diversified disciplines.

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**Example text**

N and θu (v) = v if v ∈ M − i Ui . It is clear that θu : M → M sends u01 , . . , u0n to u1 , . . , un , respectively. Observe that c−1 (u0 ) is the closed subgroup of Top(M ) consisting of all f ∈ Top(M ) such that f (u0i ) = u0i for i = 1, 2, . . , n. The formula (u, f ) → θu f deﬁnes a homeomorphism U × c−1 (u0 ) → c−1 (U ) commuting with the projections to U . The inverse homeomorphism sends any g ∈ c−1 (U ) to the pair (c(g), (θc(g) )−1 g) ∈ U × c−1 (u0 ). 36. Two elements of Top(M ) have the same image under the evaluation map e if and only if they lie in the same left coset of Top(M, Q) in Top(M ).

The ultimate goal of knot theory is a classiﬁcation of knots and links. If M has a smooth structure, then any geometric link in M is isotopic to a geometric link whose underlying 1-dimensional manifold is a smooth submanifold of M . Therefore working with links in smooth 3-dimensional manifolds, we can always restrict ourselves to smooth representatives. 2 Link diagrams The technique of braid diagrams discussed in Chapter 1 can be extended to links. We shall restrict ourselves to the case in which the ambient 3-manifold is the product of a surface Σ (possibly with boundary ∂Σ) with I.

Denote the Euclidean norm of a vector z ∈ Rn by |z|. For any self-homeomorphism h of D, the formula ht (z) = z if t ≤ |z| ≤ 1, t h(z/t) if |z| < t deﬁnes an isotopy {ht : D → D}t∈I of h0 = id to h1 = h. Note that if h(0) = 0, then ht (0) = 0 for all t ∈ I. Therefore we also have M(D, {0}) = {1}. The study of the mapping class groups leads to a vast and ramiﬁed theory; see [Iva02] for a recent survey of the mapping class groups of surfaces. We shall focus on one series of mapping class groups arising when M is a 2-disk and Q is an n-point subset of M ◦ , where n = 1, 2, .