Knotted surfaces and their diagrams / J. Scott Carter, Masahico Saito

Auteur: Carter, J. Scott (1956-) - AuteurCo-auteur: Saito, Masahico (1959-) - AuteurType de document: MonographieCollection: Mathematical surveys and monographs ; 55Langue: anglaisPays: Etats UnisÉditeur: Providence RI : American Mathematical Society, cop. 1998 Description: 1 vol. (XI-258 p.) : fig. ; 26 cm ISBN: 9780821805930 ; rel. ISSN: 0885-4653Note: Contient des exercicesRésumé: In the classical theory of knots and links in 3-space, one utilizes projections of knots and links and applies to them the Reidemeister moves, a sequence of which will take one from any one projection of a given knot or link to any other projection of that knot or link. The Reidemeister moves have played an essential role in the development of a wide variety of invariants for knots and links, since any quantity that remains unchanged by the three moves is an invariant for knots and links. In this book, the authors extend these ideas to knotted surfaces in 4-space. They begin by developing the idea of a knotted surface diagram. One projects the knotted surface to a three-dimensional hyperplane, resulting in a generic surface with self-intersections. One can then fix a height function and look at slices of the surface at non-critical levels. This generates a sequence of frames, called a movie, each frame containing a knot or link. Each frame can be described by a word, so the entire movie can be described by a sequence of words, called a sentence. One can also keep track of a projection to a 2-dimensional plane of the singular set of a knotted diagram. This is called a chart ... (MSN).Bibliographie: Bibliogr. p. 243-255. Index. Sujets MSC: 57N13 Manifolds and cell complexes -- Topological manifolds -- Topology of E4, 4-manifolds
57Q45 Manifolds and cell complexes -- PL-topology -- Knots and links (in high dimensions)
57-02 Manifolds and cell complexes -- Research exposition (monographs, survey articles)
En-ligne: MSN | zbMath
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Contient des exercices

Bibliogr. p. 243-255. Index

In the classical theory of knots and links in 3-space, one utilizes projections of knots and links and applies to them the Reidemeister moves, a sequence of which will take one from any one projection of a given knot or link to any other projection of that knot or link. The Reidemeister moves have played an essential role in the development of a wide variety of invariants for knots and links, since any quantity that remains unchanged by the three moves is an invariant for knots and links.
In this book, the authors extend these ideas to knotted surfaces in 4-space. They begin by developing the idea of a knotted surface diagram. One projects the knotted surface to a three-dimensional hyperplane, resulting in a generic surface with self-intersections. One can then fix a height function and look at slices of the surface at non-critical levels. This generates a sequence of frames, called a movie, each frame containing a knot or link. Each frame can be described by a word, so the entire movie can be described by a sequence of words, called a sentence. One can also keep track of a projection to a 2-dimensional plane of the singular set of a knotted diagram. This is called a chart ... (MSN)

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