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Graphs on Surfaces: Dualities, Polynomials, and Knots 2013 ed. [Minkštas viršelis]

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  • Formatas: Paperback / softback, 139 pages, aukštis x plotis: 235x155 mm, weight: 2409 g, 41 Illustrations, color; 41 Illustrations, black and white; XI, 139 p. 82 illus., 41 illus. in color., 1 Paperback / softback
  • Serija: SpringerBriefs in Mathematics
  • Išleidimo metai: 27-Jun-2013
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1461469708
  • ISBN-13: 9781461469704
Kitos knygos pagal šią temą:
Graphs on Surfaces: Dualities, Polynomials, and Knots 2013 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 139 pages, aukštis x plotis: 235x155 mm, weight: 2409 g, 41 Illustrations, color; 41 Illustrations, black and white; XI, 139 p. 82 illus., 41 illus. in color., 1 Paperback / softback
  • Serija: SpringerBriefs in Mathematics
  • Išleidimo metai: 27-Jun-2013
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1461469708
  • ISBN-13: 9781461469704
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781461469704.html
Raktažodžiai:

Graphs on Surfaces: Dualities, Polynomials, and Knots offers an accessible and comprehensive treatment of recent developments on generalized duals of graphs on surfaces, and their applications. The authors illustrate the interdependency between duality, medial graphs and knots; how this interdependency is reflected in algebraic invariants of graphs and knots; and how it can be exploited to solve problems in graph and knot theory. Taking a constructive approach, the authors emphasize how generalized duals and related ideas arise by localizing classical constructions, such as geometric duals and Tait graphs, and then removing artificial restrictions in these constructions to obtain full extensions of them to embedded graphs. The authors demonstrate the benefits of these generalizations to embedded graphs in chapters describing their applications to graph polynomials and knots.

Graphs on Surfaces: Dualities, Polynomials, and Knots also provides a self-contained introduction to graphs on surfaces, generalized duals, topological graph polynomials, and knot polynomials that is accessible both to graph theorists and to knot theorists. Directed at those with some familiarity with basic graph theory and knot theory, this book is appropriate for graduate students and researchers in either area. Because the area is advancing so rapidly, the authors give a comprehensive overview of the topic and include a robust bibliography, aiming to provide the reader with the necessary foundations to stay abreast of the field. The reader will come away from the text convinced of advantages of considering these higher genus analogues of constructions of plane and abstract graphs, and with a good understanding of how they arise.

Recenzijos

From the reviews:

Here, the venerable knot-theoretic and graph-theoretic themes find a host of unifying common generalizations. Undergraduates will appreciate the patient and visual development of the foundations, particularly the dualities (paired representations of a single structure). Summing Up: Recommended. Upper-division undergraduates through researchers/faculty. (D. V. Feldman, Choice, Vol. 51 (7), March, 2014)

This monograph is aimed at researchers both in graph theory and in knot theory. It should be accessible to a graduate student with a grounding in both subjects. There are (colour) diagrams throughout. The monograph gives a unified treatment of various ideas that have been studied and used previously, generalising many of them in the process. (Jessica Banks, zbMATH, Vol. 1283, 2014)

The authors have composed a very interesting and valuable work. For properly prepared readers the book under review is the occasion for all sorts of fun including the inner life of ribbon groups, Tait graphs, Penrose polynomials, Tutte polynomials, and of course Jones polynomials and HOMFLY polynomials. This is fascinating mathematics, presented in a clear and accessible way. (Michael Berg, MAA Reviews, October, 2013)

1 Embedded Graphs
1(22)
1.1 Embedded Graphs and Their Representations
1(10)
1.1.1 Abstract Graphs
1(1)
1.1.2 Surfaces
2(3)
1.1.3 Cellularly Embedded Graphs
5(1)
1.1.4 Ribbon Graphs
5(2)
1.1.5 Band Decompositions
7(1)
1.1.6 Ribbon and Arrow Marked Graphs (Ram Graphs)
8(1)
1.1.7 Arrow Presentations
9(1)
1.1.8 Signed Rotation Systems
10(1)
1.1.9 A Note on Terminology
10(1)
1.2 Further Properties of Embedded Graphs
11(3)
1.2.1 Subgraphs of Embedded Graphs
11(1)
1.2.2 Genus and Loops
12(2)
1.3 Petrials of Embedded Graphs
14(1)
1.4 Geometric Duality
14(2)
1.5 Medial Graphs, Tait Graphs, and Duality
16(7)
1.5.1 Medial Graphs
17(1)
1.5.2 Vertex States and Graph States
18(1)
1.5.3 Tait Graphs
19(4)
2 Generalised Dualities
23(20)
2.1 Partial Petrials
23(1)
2.2 Partial Duals
24(10)
2.2.1 Partial Duality with Respect to an Edge
25(2)
2.2.2 Other Constructions of Partial Duals
27(6)
2.2.3 Basic Properties of Partial Duality
33(1)
2.3 Twisted Duality
34(5)
2.3.1 Sequences of Partial Duals and Petrials
34(2)
2.3.2 Twisted Duals
36(3)
2.4 The Ribbon Group and its Action
39(4)
2.4.1 Defining the Group Action
40(1)
2.4.2 Recovering Dualities from Actions of Subgroups of the Ribbon Group
41(2)
3 Twisted Duality, Cycle Family Graphs, and Embedded Graph Equivalence
43(18)
3.1 Characterising Orb(G)
44(8)
3.1.1 Extending Tait Graphs to Cycle Family Graphs
45(2)
3.1.2 Twisted Duality and Cycle Family Graphs
47(5)
3.2 A Structural Hierarchy and Corresponding Dualities
52(6)
3.2.1 Forms of Equivalences
52(1)
3.2.2 Induced Dualities
53(5)
3.3 Properties of Some Special Orbits
58(3)
4 Interactions with Graph Polynomials
61(40)
4.1 Classical Graph Polynomials
61(2)
4.2 Deletion, Contraction, and Medial Graphs
63(2)
4.3 Twisted Duals and the Topological Transition Polynomial
65(5)
4.3.1 The Topological Transition Polynomial
66(2)
4.3.2 The Topological Transition Polynomial and the Ribbon Group Action
68(2)
4.4 The Penrose Polynomial
70(10)
4.4.1 The Penrose Polynomial of an Embedded Graph and Its Relation to the Transition Polynomial
71(2)
4.4.2 Identities for the Topological Penrose Polynomial
73(3)
4.4.3 k-Valuations and the Penrose Polynomial
76(2)
4.4.4 Graph Colouring and the Penrose Polynomial
78(2)
4.5 Topological Tutte Polynomials
80(15)
4.5.1 The Ribbon Graph Polynomial and the Topochromatic Polynomial
80(8)
4.5.2 Relation to the Topological Transition Polynomial
88(3)
4.5.3 Duality Relations for Topological Tutte Polynomials
91(1)
4.5.4 Polynomials of Signed Embedded Graphs
92(3)
4.6 Relating the Penrose and Topochromatic Polynomials
95(6)
5 Applications to Knot Theory
101(32)
5.1 Knots and Links
102(3)
5.1.1 Links in a 3-Manifold
102(1)
5.1.2 Link Diagrams
103(2)
5.2 Virtual Links
105(3)
5.2.1 Virtual Link Diagrams
105(1)
5.2.2 Virtual Links as Links in Thickened Surfaces
106(2)
5.3 Presenting Links as Embedded Graphs
108(8)
5.3.1 Signed Tait Graphs
108(3)
5.3.2 Ribbon Graphs and Link Diagrams
111(5)
5.4 The Jones Polynomial and Graph Polynomials
116(9)
5.4.1 The Jones Polynomial and the Kauffman Bracket
117(3)
5.4.2 The Jones Polynomial as a Graph Polynomial
120(3)
5.4.3 The Kauffman Bracket and the Transition Polynomial
123(2)
5.5 The HOMFLY-PT Polynomial and Graph Polynomials
125(8)
5.5.1 The HOMFLY-PT Polynomial
126(2)
5.5.2 Graph Polynomials from the HOMFLY-PT Polynomial
128(5)
References 133(4)
Index 137