Surface Knots in 4 Space An Introduction 1st edition by Seiichi Kamada ISBN 9811040907 978-9811040900

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ISBN 10: 9811040907
ISBN 13: 978-9811040900
Author: Seiichi Kamada

  This introductory volume provides the basics of surface-knots and related topics, not only for researchers in these areas but also for graduate students and researchers who are not familiar with the field.Knot theory is one of the most active research fields in modern mathematics. Knots and links are closed curves (one-dimensional manifolds) in Euclidean 3-space, and they are related to braids and 3-manifolds. These notions are generalized into higher dimensions. Surface-knots or surface-links are closed surfaces (two-dimensional manifolds) in Euclidean 4-space, which are related to two-dimensional braids and 4-manifolds. Surface-knot theory treats not only closed surfaces but also surfaces with boundaries in 4-manifolds. For example, knot concordance and knot cobordism, which are also important objects in knot theory, are surfaces in the product space of the 3-sphere and the interval.Included in this book are basics of surface-knots and the related topics ofclassical knots, the motion picture method, surface diagrams, handle surgeries, ribbon surface-knots, spinning construction, knot concordance and 4-genus, quandles and their homology theory, and two-dimensional braids.

Surface Knots in 4 Space An Introduction 1st Table of contents:

1 Surface-Knots

1.1 Preliminaries

1.2 Surface-Knots

1.3 Orientations of Surface-Knots and the Ambient Space

1.4 Non-orientable Surface-Knots

1.5 Surface-Knots in the 4-Sphere

1.6 PL Surface-Knots

2 Knots

2.1 Knots and Diagrams

2.2 Seifert Surfaces

2.3 Meridians and Longitudes

2.4 Band Surgeries and Connected Sums

2.5 Knot Groups

2.6 Seifert Matrices

2.7 Skein Relations and Polynomial Invariants

2.8 2-Bridge Knots, Torus Knots, Satellite Knots

3 Motion Pictures

3.1 Motion Pictures

3.2 Normal Forms of Surface-Knots

3.3 Trivial Disk Systems

3.4 Link Transformation Sequences

3.5 Links with Bands

3.6 ch-Diagrams

3.7 Normal Euler Number

3.8 Knot Groups and Elementary Ideals

4 Surface Diagrams

4.1 Surface Diagrams

4.2 Roseman Moves

4.3 Computation of the Surface-Knot Group from a Diagram

4.4 Diagrams and Normal Euler Numbers

4.5 The Triple Point Number and the Sheet Number

4.6 The Triple Linking Number

5 Handle Surgery and Ribbon Surface-Knots

5.1 1-Handles

5.2 Classifying 1-Handles

5.3 2-Handles

5.4 Handle Sum and Connected Sum

5.5 Ribbon Knots

5.6 Ribbon Surface-Knots

5.7 Unknotting Surface-Links by 1-Handle Surgery

6 Spinning Construction

6.1 Spinning Construction

6.2 Deform-Spinning 1

6.3 Deform-Spinning 2

6.4 Spinning Construction for P2-Knots

6.5 Meridians of P2-Knots

7 Knot Concordance

7.1 Slice Knots

7.2 Knot Concordance

7.3 Concordance and Cobordism on Links

7.4 The 4-Genus

8 Quandles

8.1 Fox’s Coloring

8.2 Keis

8.3 Quandles

8.4 Quandle Colorings

8.5 Fenn and Rourke’s Notation

8.6 Presentations of a Rack and a Quandle

8.7 Presentations of a Rack and a Quandle, 2

8.8 Associated Groups of Quandles

8.9 Knot Quandles

9 Quandle Homology Groups and Invariants

9.1 Quandle Homology Groups

9.2 Quandle Cocycle Invariants of Knots

9.3 Quandle Cocycle Invariants of Surface-Knots

9.4 Quandle Cocycle Invariants with Region Colorings

9.5 Symmetric Quandles

10 2-Dimensional Braids

10.1 Braids and Knots

10.2 2-Dimensional Braids

10.3 Motion Pictures

10.4 Monodromies

10.5 Chart Descriptions

10.6 Braid Presentation of Surface-Links

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Tags: Seiichi Kamada, Surface Knots, An Introduction