Knots and Surfaces

by Gilbert, N. D.; Porter, T.

• ISBN13: 9780198514909
• ISBN10: 0198514905
• Edition: Reprint
• Format: Paperback
• Copyright: 1996-05-23
• Publisher: Oxford University Press

Summary

This highly readable text details the interaction between the mathematical theory of knots and the theories of surfaces and group presentations. It expertly introduces several topics critical to the development of pure mathematics while providing an account of math "in action" in an unusual context. Beginning with a simple diagrammatic approach to the study of knots that reflects the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent research advances. Topics include topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. The authors skillfully combine these topics to form a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces to students and researchers in mathematics.

Table of Contents

Knots, links, and diagrams	1	(18)
Knot and link diagrams	1	(4)
Isotopy and the Reidemeister moves	5	(3)
3-colouring	8	(3)
Numerical invariants	11	(4)
Chiral and invertible knots	15	(4)
Knot and link polynomials	19	(23)
State models and the Jones polynomial	19	(9)
The Jones polynomial	28	(2)
Applications of the Kauffman polynomial	30	(4)
The oriented or Homfly polynomial	34	(4)
The Alexander polynomial	38	(4)
Topological spaces	42	(13)
Topological spaces	42	(3)
Continuity	45	(3)
Connectedness	48	(1)
Identification spaces	49	(2)
Knots and isotopy	51	(4)
Surfaces	55	(41)
Combinatorial surfaces	57	(11)
Cutting and pasting	68	(12)
Invariants of surfaces: orientability and boundary curves	80	(4)
Invariants of surfaces: Euler characteristic and genus	84	(6)
Knots and surfaces	90	(6)
The arithmetic of knots	96	(15)
The sum of oriented knots	96	(4)
Genus of K + L	100	(3)
Bridge number of K + L	103	(2)
Factorizations, prime knots, and the factorization theorem	105	(2)
n-colourability	107	(1)
Polynomials and the sum of knots	108	(3)
Presentations of groups	111	(52)
Examples of presentations	111	(4)
Dihedral groups	115	(3)
Cayley quivers	118	(4)
Free groups	122	(7)
Presentations	129	(2)
Tietze transformations	131	(7)
Quotient groups	138	(1)
Abelianization	139	(1)
Finitely generated Abelian groups	140	(4)
Pushouts of groups	144	(6)
Knot groups	150	(13)
Graphs and trees	163	(15)
Graphs, quivers, and trees	163	(6)
Examples of graphs	169	(2)
Planar graphs	171	(2)
Embedding graphs in surfaces	173	(5)
Alexander matrices and Alexander polynomials	178	(19)
Group rings	178	(4)
Derivatives	182	(2)
The Alexander matrix	184	(3)
Elementary ideals	187	(5)
Alexander polynomials via the Wirtinger presentation	192	(5)
The fundamental group	197	(15)
Paths	197	(2)
Homotopy	199	(4)
Calculations of the fundamental group	203	(5)
Homotopy equivalence	208	(4)
Van Kampen's theorem	212	(12)
Applications of the Van Kampen theorem	224	(26)
Calculations of fundamental groups	224	(9)
Surface groups	233	(3)
Spaces with finite cyclic fundamental group	236	(1)
Spaces with given fundamental group	237	(2)
The group of a knot	239	(4)
Torus knots	243	(3)
Analysis of the knot group Gm,n	246	(4)
Covering spaces	250	(14)
Covering maps and covering spaces	250	(8)
Deck transformations and universal covers	258	(1)
Covering graphs	259	(5)
Bibliography	264	(2)
Index	266