Book contents
- Frontmatter
- ACKNOWLEDGEMENTS
- PREFACE
- Contents
- Chapter 1 A CENTURY OF KNOT THEORY
- Chapter 2 WHAT IS A KNOT?
- Chapter 3 COMBINATORIAL TECHNIQUES
- Chapter 4 GEOMETRIC TECHNIQUES
- Chapter 5 ALGEBRAIC TECHNIQUES
- Chapter 6 GEOMETRY, ALGEBRA, AND THE ALEXANDER POLYNOMIAL
- Chapter 7 NUMERICAL INVARIANTS
- Chapter 8 SYMMETRIES OF KNOTS
- Chapter 9 HIGH-DIMENSIONAL KNOT THEORY
- Chapter 10 NEW COMBINATORIAL TECHNIQUES
- Appendix 1 KNOT TABLE
- Appendix 2 ALEXANDER POLYNOMIALS
- REFERENCES
- INDEX
Chapter 9 - HIGH-DIMENSIONAL KNOT THEORY
- Frontmatter
- ACKNOWLEDGEMENTS
- PREFACE
- Contents
- Chapter 1 A CENTURY OF KNOT THEORY
- Chapter 2 WHAT IS A KNOT?
- Chapter 3 COMBINATORIAL TECHNIQUES
- Chapter 4 GEOMETRIC TECHNIQUES
- Chapter 5 ALGEBRAIC TECHNIQUES
- Chapter 6 GEOMETRY, ALGEBRA, AND THE ALEXANDER POLYNOMIAL
- Chapter 7 NUMERICAL INVARIANTS
- Chapter 8 SYMMETRIES OF KNOTS
- Chapter 9 HIGH-DIMENSIONAL KNOT THEORY
- Chapter 10 NEW COMBINATORIAL TECHNIQUES
- Appendix 1 KNOT TABLE
- Appendix 2 ALEXANDER POLYNOMIALS
- REFERENCES
- INDEX
Summary
The theory of knots in R3 naturally generalizes to a study of knotting in Rn, with n > 3, and many new and fascinating aspects of knot theory appear in this high-dimensional setting. What is perhaps most surprising is that many problems that are intractable in the classical case have been solved for high-dimensional knots. There is also a strong interplay between knot theory in different dimensions, and this interplay leads to an array of new topics at the border of the classical and high-dimensional settings.
The definitions of polygonal knot and of deformation of knots generalizes immediately to R4, (or for that matter Rn); one can simply consider sequences of points in 4-space instead of R3. Knots formed in this way are called 1- dimensional knots in 4-space, or, more briefly, 1-knots. It turns out though that there is really no interesting theory of such 1-knots; all such knots in 4-space are equivalent.
The appropriate generalization increases the dimension of the knot as well as the dimension of the ambient space. The definition of surface given in Chapter 4 easily generalizes to yield a definition of surfaces in 4-space. A 2-knot is a surface in R4 that is homeomorphic to S2, the standard sphere in 3-space. Figure 9.1 is a schematic illustration of such a knot. Section 1 discusses some of the details of the definitions, as well as a new subtlety that arises at the foundation of the subject.
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- Knot Theory , pp. 179 - 204Publisher: Mathematical Association of AmericaPrint publication year: 1993