By Kunio Murasugi, B. Kurpita

This ebook offers a complete exposition of the speculation of braids, starting with the fundamental mathematical definitions and buildings. one of the issues defined intimately are: the braid team for numerous surfaces; the answer of the note challenge for the braid staff; braids within the context of knots and hyperlinks (Alexander's theorem); Markov's theorem and its use in acquiring braid invariants; the relationship among the Platonic solids (regular polyhedra) and braids; using braids within the answer of algebraic equations. Dirac's challenge and certain forms of braids termed Mexican plaits are additionally mentioned. viewers: because the publication is dependent upon suggestions and strategies from algebra and topology, the authors additionally offer a few appendices that disguise the required fabric from those branches of arithmetic. therefore, the ebook is on the market not just to mathematicians but additionally to anyone who may have an curiosity within the conception of braids. specifically, as increasingly more functions of braid idea are stumbled on outdoors the world of arithmetic, this e-book is perfect for any physicist, chemist or biologist who want to comprehend the arithmetic of braids. With its use of diverse figures to provide an explanation for truly the math, and routines to solidify the certainty, this e-book can also be used as a textbook for a direction on knots and braids, or as a supplementary textbook for a direction on topology or algebra.

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**Example text**

If S has nonempty boundary and has a hyperbolic metric, then S is isometric to a totally geodesic subspace of H2 . Similarly, if S has a Euclidean metric, then S is isometric to a totally geodesic subspace of the Euclidean plane E2 . 2 Let S be any surface (perhaps with punctures or boundary). If χ(S) < 0, then S admits a hyperbolic metric. If χ(S) = 0, then S admits a Euclidean metric. A surface endowed with a ﬁxed hyperbolic metric will be called a hyperbolic surface. A surface with a Euclidean metric will be called a Euclidean surface or ﬂat surface.

13 gives us a way to replace homeomorphisms with diffeomorphisms. We can also replace isotopies with smooth isotopies. In other words, if two diffeomorphisms are isotopic, then they are smoothly isotopic; see, for example, [30]. In this book, we will switch between the topological setting and the smooth setting as is convenient. For example, when deﬁning a map of a surface to itself (either by equations or by pictures), it is often easier to write down a homeomorphism than a smooth map. On the other hand, when we need to appeal to transversality, extension of isotopy, and so on, we will need to assume we have a diffeomorphism.

There is then a homotopy from γ to the standard straight-line representative of (p, q) ∈ π1 (T 2 ); indeed, the straight-line homotopy from the lift of γ to the straight line through (0, 0) and (p, q) is equivariant with respect to the group of deck transformations and thus descends to the desired homotopy. Now, if a closed curve in T 2 is simple, then its straight-line representative is simple. Thus we have the following fact. 5 The nontrivial homotopy classes of oriented simple closed curves in T 2 are in bijective correspondence with the set of primitive elements of π1 (T 2 ) ≈ Z2 .