An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces (Crc Press Series on Discrete Mathematics and Its Applications)

Maps are beguilingly simple structures with deep and ubiquitous properties. They arise in an essential way in many areas of mathematics and mathematical physics, but require considerable time and computational effort to generate. Few collected drawings are available for reference, and little has been written, in book form, about their enumerative aspects. An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces is the first book to provide complete collections of maps along with their vertex and face partitions, number of rootings, and an index number for cross referencing. It provides an explanation of axiomatization and encoding, and serves as an introduction to maps as a combinatorial structure. The Atlas lists the maps first by genus and number of edges, and gives the embeddings of all graphs with at most five edges in orientable surfaces, thus presenting the genus distribution for each graph. Exemplifying the use of the Atlas, the authors explore two substantial conjectures with origins in mathematical physics and geometry: the Quadrangulation Conjecture and the b-Conjecture. The authors' clear, readable exposition and overview of enumerative theory makes this collection accessible even to professionals who are not specialists. For researchers and students working with maps, the Atlas provides a ready source of data for testing conjectures and exploring the algorithmic and algebraic properties of maps publisher's description An Atlas Of The Smaller Maps In Orientable And Nonorientable Surfaces Is The First Book To Provide Complete Collections Of Maps Along With Their Vertex And Face Partitions, Number Of Rootings, And An Index Number For Cross-referencing. It Provides An Explanation Of Axiomatization And Encoding, And Serves As An Introduction To Maps As A Combinatorial Structure. The Atlas Lists The Maps First By Genus And Number Of Edges, And Gives The Embeddings Of All Graphs With At Most Five Edges In Orientable Surfaces, Thus Presenting The Genus Distribution For Each Graph.--jacket. 2 Surfaces And Maps 9 -- 2.1.2 Polygonal Representation Of Orientable Surfaces 11 -- 2.1.3 Polygonal Representation Of Nonorientable Surfaces 12 -- 2.1.4 Rooting, Associated Graph And Dual 14 -- 2.1.5 The Entry For A Map In The Atlas 17 -- 2.3.1 Example: Do The Vertex And Face Partitions Determine The Number Of Rootings? 22 -- 2.3.2 Example: Maps With The Same Vertex And Face Partitions But Different Associated Graphs 23 -- 2.3.3 Examples Of Nonrealizability 23 -- 2.4 An Application Of [kappa]-realizable Partitions 25 -- 2.4.1 The Absolute Galois Group 25 -- 2.4.2 Belyi Functions 25 -- 3 The Axiomatization And The Encoding Of Maps 29 -- 3.1 Orientable Surfaces 29 -- 3.1.1 Axiomatization For Maps In Orientable Surfaces 29 -- 3.1.2 Encoding A Map As A Permutation 30 -- 3.1.3 Construction Of The Set Of All Rooted Maps 31 -- 3.1.4 Determining The Number Of Rootings 37 -- 3.2 Locally Orientable Surfaces 38 -- 3.2.1 Axiomatization For Maps In Locally Orientable Surfaces 38 -- 3.2.2 Encoding A Map As A Permutation 40 -- 3.2.3 Constructing The Set Of All Rooted Maps 41 -- 4 Generating Series And Conjectures 47 -- 4.1 Generating Series For Hypermaps 47 -- 4.1.1 Schur Functions And Zonal Polynomials 47 -- 4.1.2 Genus Series For Rooted Hypermaps 48 -- 4.1.3 Two Algebras 50 -- 4.2 Specialization To Maps 51 -- 4.2.1 The Genus Series For Maps In Orientable Surfaces 51 -- 4.2.2 The Genus Series For Maps In Locally Orientable Surfaces 52 -- 4.3 The Quadrangulation Conjecture 53 -- 4.3.1 An Informal Principle Of Enumerative Combinatorics 53 -- 4.3.4 Generalization To Eulerian Maps And The Bijection [omega] 58 -- 4.3.5 Setwise Action Of The Bijection [omega] 60 -- 4.4.2 Jack Symmetric Functions 65 -- 5 Maps In Orientable Surfaces 73 -- 5.1 Genus 0 -- The Sphere 73 -- 5.2 Genus 1 -- The Torus 88 -- 5.3 Genus 2 -- The Double Torus 107 -- 6 Maps In Nonorientable Surfaces 115 -- 6.1 Genus 1 -- The Projective Plane 115 -- 6.2 Genus 2 -- The Klein Bottle 121 -- 6.3 Genus 3 -- The Crosscapped Torus 127 -- 6.4 Genus 4 -- The Doubly Crosscapped Torus 135 -- 7 Face Regular Maps And Hypermaps 139 -- 7.1 Triangulations 139 -- 7.1.1 Orientable Of Genus 0 And 1 139 -- 7.1.2 Nonorientable Of Genus 1 And 2 140 -- 7.2 Quadrangulations 141 -- 7.2.1 Orientable Of Genus 0, 1 And 2 141 -- 7.2.2 Nonorientable Of Genus 1, 2 And 3 143 -- 7.3 Hypermaps 145 -- 7.3.1 Orientable Of Genus 0, 1 And 2 146 -- 7.3.2 Nonorientable Of Genus 1, 2 And 3 151 -- 8 Associated Graphs And Their Maps 153 -- 9 Numbers Of Rooted Maps 211 -- 9.1 Orientable: By Vertex And Face Partition 211 -- 9.2 Nonorientable: By Vertex And Face Partition 218 -- 9.3 Summarized By Edges And Vertices 227 -- 9.3.1 All Maps By Number Of Edges 227 -- 9.3.2 All Maps By Numbers Of Edges And Vertices 228 -- 10 Numbers Of Unrooted Maps 231 -- 10.1 Orientable: By Vertex And Face Partition 231 -- 10.2 Nonorientable: By Vertex And Face Partition 234 -- 11 Nonrealizable Pairs Of Partitions 239 -- 11.1 For Orientable Surfaces 239 -- 11.2 For Nonorientable Surfaces 243 -- 12 Map Polynomials 245 -- 12.1 B-polynomials 245 -- 12.1.1 Hypermaps 245 -- 12.2 Genus Distributions 264. David M. Jackson And Terry I. Visentin. Includes Bibliographical References (p. 271-273) And Index. "An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces is the first book to provide complete collections of maps along with their vertex and face partitions, number of rootings, and an index number for cross-referencing. It provides an explanation of axiomatization and encoding, and serves as an introduction to maps as a combinatorial structure. The Atlas lists the maps first by genus and number of edges, and gives the embeddings of all graphs with at most five edges in orientable surfaces, thus presenting the genus distribution for each graph."--Résumé de l'éditeur &Quot;An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces is the first book to provide complete collections of maps along with their vertex and face partitions, number of rootings, and an index number for cross-referencing. It provides an explanation of axiomatization and encoding, and serves as an introduction to maps as a combinatorial structure. The Atlas lists the maps first by genus and number of edges, and gives the embeddings of all graphs with at most five edges in orientable surfaces, thus presenting the genus distribution for each graph."--BOOK JACKET This text provides a complete collection of drawings of smaller maps along with their number of rootings and serves as an introduction to maps as a combinatorial structure. It presents an introduction to the enumerative theory of maps and gives an explanation of their axiomatization and encoding This Atlas gives a complete listing of maps and hypermaps with a small number of edges for both orientable and nonorientable surfaces, and the numbers of their rootings.