Verfügbarkeit: Graphs on surfaces and their applications

Graphs on surfaces and their applications:

Gespeichert in:

Bibliographische Detailangaben
Hauptverfasser:	Lando, Sergej K. 1955- (VerfasserIn), Zvonkin, Aleksandr K. 1948- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer 2004
Schriftenreihe:	Encyclopaedia of mathematical sciences 141
Schlagworte:	Topologische Graphentheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Literaturverz. S. 430 - 444
Beschreibung:	XV, 455 S. graph. Darst.
ISBN:	3540002030

Internformat

MARC


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100	1		\|a Lando, Sergej K. \|d 1955- \|e Verfasser \|0 (DE-588)128627158 \|4 aut
245	1	0	\|a Graphs on surfaces and their applications \|c Sergei K. Lando ; Alexander K. Zvonkin. Appendix by Don B. Zagier
264		1	\|a Berlin [u.a.] \|b Springer \|c 2004
300			\|a XV, 455 S. \|b graph. Darst.
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490	1		\|a Encyclopaedia of mathematical sciences \|v 141
490	1		\|a Encyclopaedia of mathematical sciences / Low-dimensional topology \|v 2
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700	1		\|a Zvonkin, Aleksandr K. \|d 1948- \|e Verfasser \|0 (DE-588)128627166 \|4 aut
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Datensatz im Suchindex

_version_	1804129839343468544
adam_text	CONTENTS 0 INTRODUCTION: WHAT IS THIS BOOK ABOUT . . . . . . . . . . . . . . . . . . . 1 0.1 NEW LIFE OF AN OLD THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 PLAN OF THE BOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 WHAT YOU WILL NOT FIND IN THIS BOOK . . . . . . . . . . . . . . . . . . . . . 4 1 CONSTELLATIONS, COVERINGS, AND MAPS . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 CONSTELLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 RAMIFIED COVERINGS OF THE SPHERE . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 FIRST DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.2 COVERINGS AND FUNDAMENTAL GROUPS . . . . . . . . . . . . . . . . . 15 1.2.3 RAMIFIED COVERINGS OF THE SPHERE AND CONSTELLATIONS . . . 18 1.2.4 SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.1 GRAPHS VERSUS MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 MAPS: TOPOLOGICAL DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.3 MAPS: PERMUTATIONAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4 CARTOGRAPHIC GROUPS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.5 HYPERMAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.5.1 HYPERMAPS AND BIPARTITE MAPS . . . . . . . . . . . . . . . . . . . . . 43 1.5.2 TREES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.5.3 APPENDIX: FINITE LINEAR GROUPS . . . . . . . . . . . . . . . . . . . . . 49 1.5.4 CANONICAL TRIANGULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.6 MORE THAN THREE PERMUTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.6.1 PREIMAGES OF A STAR OR OF A POLYGON. . . . . . . . . . . . . . . . . . 56 1.6.2 CACTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.6.3 PREIMAGES OF A JORDAN CURVE . . . . . . . . . . . . . . . . . . . . . . . 61 1.7 FURTHER DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1.7.1 COVERINGS OF SURFACES OF HIGHER GENERA. . . . . . . . . . . . . . . 63 1.7.2 RITTS THEOREM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.7.3 SYMMETRIC AND REGULAR CONSTELLATIONS . . . . . . . . . . . . . . . 68 1.8 REVIEW OF RIEMANN SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 XII CONTENTS 2 DESSINS DENFANTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 INTRODUCTION: THE BELYI THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 PLANE TREES AND SHABAT POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . 80 2.2.1 GENERAL THEORY APPLIED TO TREES . . . . . . . . . . . . . . . . . . . . 80 2.2.2 SIMPLE EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.3 FURTHER DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.2.4 MORE ADVANCED EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.3 BELYI FUNCTIONS AND BELYI PAIRS . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.4 GALOIS ACTION AND ITS COMBINATORIAL INVARIANTS . . . . . . . . . . . . . . 115 2.4.1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.4.2 GALOIS INVARIANTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.4.3 TWO THEOREMS ON TREES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.5 SEVERAL FACETS OF BELYI FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2.5.1 A BOUND OF DAVENPORTSTOTHERSZANNIER . . . . . . . . . . . . . 126 2.5.2 JACOBI POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.5.3 FERMAT CURVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.5.4 THE ABC CONJECTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.5.5 JULIA SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2.5.6 PELL EQUATION FOR POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . 142 2.6 PROOF OF THE BELYI THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 2.6.1 THE ONLY IF PART OF THE BELYI THEOREM. . . . . . . . . . . . . 146 2.6.2 COMMENTS TO THE PROOF OF THE ONLY IF PART . . . . . . . . . 147 2.6.3 THE IF, OR THE OBVIOUS PART OF THE BELYI THEOREM . 150 3 INTRODUCTION TO THE MATRIX INTEGRALS METHOD . . . . . . . . . . . . . . 155 3.1 MODEL PROBLEM: ONE-FACE MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.2 GAUSSIAN INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2.1 THE GAUSSIAN MEASURE ON THE LINE . . . . . . . . . . . . . . . . . . 160 3.2.2 GAUSSIAN MEASURES IN R K . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.2.3 INTEGRALS OF POLYNOMIALS AND THE WICK FORMULA . . . . . . . 163 3.2.4 A GAUSSIAN MEASURE ON THE SPACE OF HERMITIAN MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.2.5 MATRIX INTEGRALS AND POLYGON GLUINGS . . . . . . . . . . . . . . . . 167 3.2.6 COMPUTING GAUSSIAN INTEGRALS. UNITARY INVARIANCE . . . . . 171 3.2.7 COMPUTATION OF THE INTEGRAL FOR ONE FACE GLUINGS . . . . . 176 3.3 MATRIX INTEGRALS FOR MULTI-FACED MAPS . . . . . . . . . . . . . . . . . . . . . 179 3.3.1 FEYNMAN DIAGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 3.3.2 THE MATRIX INTEGRAL FOR AN ARBITRARY GLUING . . . . . . . . . . 180 3.3.3 GETTING RID OF DISCONNECTED GRAPHS . . . . . . . . . . . . . . . . . 183 3.4 ENUMERATION OF COLORED GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.4.1 TWO-MATRIX INTEGRALS AND THE ISING MODEL . . . . . . . . . . . . 185 3.4.2 THE GAUSS PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.4.3 MEANDERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 3.4.4 ON ENUMERATION OF MEANDERS . . . . . . . . . . . . . . . . . . . . . . . 191 3.5 COMPUTATION OF MATRIX INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . 192 CONTENTS XIII 3.5.1 EXAMPLE: COMPUTING THE VOLUME OF THE UNITARY GROUP 192 3.5.2 GENERALIZED HERMITE POLYNOMIALS . . . . . . . . . . . . . . . . . . . . 195 3.5.3 PLANAR APPROXIMATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.6 KORTEWEGDE VRIES (KDV) HIERARCHY FOR THE UNIVERSAL ONE-MATRIX MODEL . . . . . . . . . . . . . . . . . . . . . . . 199 3.6.1 SINGULAR BEHAVIOR OF GENERATING FUNCTIONS . . . . . . . . . . . 200 3.6.2 THE OPERATOR OF MULTIPLICATION BY IN THE DOUBLE SCALING LIMIT. . . . . . . . . . . . . . . . . . . . . . . . . 202 3.6.3 THE ONE-MATRIX MODEL AND THE KDV HIERARCHY . . . . . . . 204 3.6.4 CONSTRUCTING SOLUTIONS TO THE KDV HIERARCHY FROM THE SATO GRASSMANIAN . . . . . . . . . . . . . . . . . . . . . . . . . 206 3.7 PHYSICAL INTERPRETATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3.7.1 MATHEMATICAL RELATIONS BETWEEN PHYSICAL MODELS . . . . . 211 3.7.2 FEYNMAN PATH INTEGRALS AND STRING THEORY . . . . . . . . . . . 211 3.7.3 QUANTUM FIELD THEORY MODELS . . . . . . . . . . . . . . . . . . . . . . 213 3.7.4 OTHER MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.8 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 3.8.1 GENERATING FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 3.8.2 CONNECTED AND DISCONNECTED OBJECTS . . . . . . . . . . . . . . . . 217 3.8.3 LOGARITHM OF A POWER SERIES AND WICKS FORMULA . . . . . . 219 4 GEOMETRY OF MODULI SPACES OF COMPLEX CURVES . . . . . . . . . . . 223 4.1 GENERALITIES ON NODAL CURVES AND ORBIFOLDS . . . . . . . . . . . . . . . . . 223 4.1.1 DIFFERENTIALS AND NODAL CURVES . . . . . . . . . . . . . . . . . . . . . . 223 4.1.2 QUADRATIC DIFFERENTIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.1.3 ORBIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.2 MODULI SPACES OF COMPLEX STRUCTURES . . . . . . . . . . . . . . . . . . . . . . 232 4.3 THE DELIGNEMUMFORD COMPACTIFICATION . . . . . . . . . . . . . . . . . . . . 234 4.4 COMBINATORIAL MODELS OF THE MODULI SPACES OF CURVES. . . . . . . . 237 4.5 ORBIFOLD EULER CHARACTERISTIC OF THE MODULI SPACES . . . . . . . . . . 243 4.6 INTERSECTION INDICES ON MODULI SPACES AND THE STRING AND DILATON EQUATIONS . . . . . . . . . . . . . . . . . . . . . . 249 4.7 KDV HIERARCHY AND WITTENS CONJECTURE . . . . . . . . . . . . . . . . . . . 256 4.8 THE KONTSEVICH MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4.9 A SKETCH OF KONTSEVICHS PROOF OF WITTENS CONJECTURE . . . . . . . 263 4.9.1 THE GENERATING FUNCTION FOR THE KONTSEVICH MODEL . . . . 263 4.9.2 THE KONTSEVICH MODEL AND INTERSECTION THEORY . . . . . . . 264 4.9.3 THE KONTSEVICH MODEL AND THE KDV EQUATION . . . . . . . . 266 5 MEROMORPHIC FUNCTIONS AND EMBEDDED GRAPHS . . . . . . . . . . . . 269 5.1 THE LYASHKOLOOIJENGA MAPPING AND RIGID CLASSIFICATION OF GENERIC POLYNOMIALS . . . . . . . . . . . . . 270 5.1.1 THE LYASHKOLOOIJENGA MAPPING . . . . . . . . . . . . . . . . . . . . 270 5.1.2 CONSTRUCTION OF THE LL MAPPING ON THE SPACE OF GENERIC POLYNOMIALS . . . . . . . . . . . . . . . . . 271 XIV CONTENTS 5.1.3 PROOF OF THE LYASHKOLOOIJENGA THEOREM . . . . . . . . . . . . . 273 5.2 RIGID CLASSIFICATION OF NONGENERIC POLYNOMIALS AND THE GEOMETRY OF THE DISCRIMINANT . . . . . . . . . . . . . . . . . . . . . 277 5.2.1 THE DISCRIMINANT IN THE SPACE OF POLYNOMIALS AND ITS STRATIFICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5.2.2 STATEMENT OF THE ENUMERATION THEOREM . . . . . . . . . . . . . . 279 5.2.3 PRIMITIVE STRATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 5.2.4 PROOF OF THE ENUMERATION THEOREM . . . . . . . . . . . . . . . . . . 282 5.3 RIGID CLASSIFICATION OF GENERIC MEROMORPHIC FUNCTIONS AND GEOMETRY OF MODULI SPACES OF CURVES . . . . . . . . . . . . . . . . . . 288 5.3.1 STATEMENT OF THE ENUMERATION THEOREM . . . . . . . . . . . . . . 288 5.3.2 CALCULATIONS: GENUS 0 AND GENUS 1 . . . . . . . . . . . . . . . . . . 289 5.3.3 CONES AND THEIR SEGRE CLASSES . . . . . . . . . . . . . . . . . . . . . . 292 5.3.4 CONES OF PRINCIPAL PARTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 5.3.5 HURWITZ SPACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 5.3.6 COMPLETED HURWITZ SPACES AND STABLE MAPPINGS . . . . . . 299 5.3.7 EXTENDING THE LL MAPPING TO COMPLETED HURWITZ SPACES . . . . . . . . . . . . . . . . . . . . . . . 300 5.3.8 COMPUTING THE TOP SEGRE CLASS; END OF THE PROOF . . . . . 302 5.4 THE BRAID GROUP ACTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 5.4.1 BRAID GROUPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 5.4.2 BRAID GROUP ACTION ON CACTI: GENERALITIES . . . . . . . . . . . . 309 5.4.3 EXPERIMENTAL STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 5.4.4 PRIMITIVE AND IMPRIMITIVE MONODROMY GROUPS . . . . . . . . 318 5.4.5 PERSPECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 5.5 MEGAMAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 5.5.1 HURWITZ SPACES OF COVERINGS WITH FOUR RAMIFICATION POINTS . . . . . . . . . . . . . . . . . . . . . . 328 5.5.2 REPRESENTATION OF H AS A DESSIN DENFANT . . . . . . . . . . . . 329 5.5.3 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 6 ALGEBRAIC STRUCTURES ASSOCIATED WITH EMBEDDED GRAPHS .. 337 6.1 THE BIALGEBRA OF CHORD DIAGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.1.1 CHORD DIAGRAMS AND ARC DIAGRAMS . . . . . . . . . . . . . . . . . . 337 6.1.2 THE 4-TERM RELATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 6.1.3 MULTIPLYING CHORD DIAGRAMS. . . . . . . . . . . . . . . . . . . . . . . . 342 6.1.4 A BIALGEBRA STRUCTURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 6.1.5 STRUCTURE THEOREM FOR THE BIALGEBRA M . . . . . . . . . . . . . . 346 6.1.6 PRIMITIVE ELEMENTS OF THE BIALGEBRA OF CHORD DIAGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6.2 KNOT INVARIANTS AND ORIGINS OF CHORD DIAGRAMS . . . . . . . . . . . . . 350 6.2.1 KNOT INVARIANTS AND THEIR EXTENSION TO SINGULAR KNOTS . 350 6.2.2 INVARIANTS OF FINITE ORDER . . . . . . . . . . . . . . . . . . . . . . . . . . 353 6.2.3 DEDUCING 1-TERM AND 4-TERM RELATIONS FOR INVARIANTS . . 355 CONTENTS XV 6.2.4 CHORD DIAGRAMS OF SINGULAR LINKS . . . . . . . . . . . . . . . . . . . 357 6.3 WEIGHT SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 6.3.1 A BIALGEBRA STRUCTURE ON THE MODULE V OF VASSILIEV KNOT INVARIANTS . . . . . . . . . . . . . . . . . . . . . . . . . 359 6.3.2 RENORMALIZATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 6.3.3 WEIGHT SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 6.3.4 VASSILIEV KNOT INVARIANTS AND OTHER KNOT INVARIANTS . . . 364 6.4 CONSTRUCTING WEIGHT SYSTEMS VIA INTERSECTION GRAPHS . . . . . . . . 367 6.4.1 THE INTERSECTION GRAPH OF A CHORD DIAGRAM . . . . . . . . . . 367 6.4.2 TUTTE FUNCTIONS FOR GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . 368 6.4.3 THE 4-BIALGEBRA OF GRAPHS. . . . . . . . . . . . . . . . . . . . . . . . . . 369 6.4.4 THE BIALGEBRA OF WEIGHTED GRAPHS . . . . . . . . . . . . . . . . . . 379 6.4.5 CONSTRUCTING VASSILIEV INVARIANTS FROM 4-INVARIANTS . . . . 383 6.5 CONSTRUCTING WEIGHT SYSTEMS VIA LIE ALGEBRAS . . . . . . . . . . . . . . 384 6.5.1 FREE ASSOCIATIVE ALGEBRAS . . . . . . . . . . . . . . . . . . . . . . . . . . 385 6.5.2 UNIVERSAL ENVELOPING ALGEBRAS OF LIE ALGEBRAS . . . . . . . . 387 6.5.3 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 6.6 SOME OTHER ALGEBRAS OF EMBEDDED GRAPHS . . . . . . . . . . . . . . . . . 393 6.6.1 CIRCLE DIAGRAMS AND OPEN DIAGRAMS. . . . . . . . . . . . . . . . . 393 6.6.2 THE ALGEBRA OF 3-GRAPHS . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 6.6.3 THE TEMPERLEYLIEB ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . 395 A APPLICATIONS OF THE REPRESENTATION THEORY OF FINITE GROUPS ( BY DON ZAGIER ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 A.1 REPRESENTATION THEORY OF FINITE GROUPS . . . . . . . . . . . . . . . . . . . . 399 A.1.1 IRREDUCIBLE REPRESENTATIONS AND CHARACTERS . . . . . . . . . . . 399 A.1.2 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 A.1.3 FROBENIUSS FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 A.2 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 A.2.1 REPRESENTATIONS OF S N AND CANONICAL POLYNOMIALS ASSOCIATED TO PARTITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A.2.2 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 A.2.3 FIRST APPLICATION: ENUMERATION OF POLYGON GLUINGS . . . . 416 A.2.4 SECOND APPLICATION: THE GOULDENJACKSON FORMULA . . . . 418 A.2.5 THIRD APPLICATION: MIRROR SYMMETRY IN DIMENSION ONE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
any_adam_object	1
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spelling	Lando, Sergej K. 1955- Verfasser (DE-588)128627158 aut Graphs on surfaces and their applications Sergei K. Lando ; Alexander K. Zvonkin. Appendix by Don B. Zagier Berlin [u.a.] Springer 2004 XV, 455 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 141 Encyclopaedia of mathematical sciences / Low-dimensional topology 2 Literaturverz. S. 430 - 444 Topologische Graphentheorie (DE-588)4341226-9 gnd rswk-swf Topologische Graphentheorie (DE-588)4341226-9 s DE-604 Zvonkin, Aleksandr K. 1948- Verfasser (DE-588)128627166 aut Zagier, Don 1951- Sonstige (DE-588)120415569 oth Low-dimensional topology Encyclopaedia of mathematical sciences 2 (DE-604)BV014813608 2 Encyclopaedia of mathematical sciences 141 (DE-604)BV024126459 141 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010211596&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Lando, Sergej K. 1955- Zvonkin, Aleksandr K. 1948- Graphs on surfaces and their applications Encyclopaedia of mathematical sciences Topologische Graphentheorie (DE-588)4341226-9 gnd
subject_GND	(DE-588)4341226-9
title	Graphs on surfaces and their applications
title_auth	Graphs on surfaces and their applications
title_exact_search	Graphs on surfaces and their applications
title_full	Graphs on surfaces and their applications Sergei K. Lando ; Alexander K. Zvonkin. Appendix by Don B. Zagier
title_fullStr	Graphs on surfaces and their applications Sergei K. Lando ; Alexander K. Zvonkin. Appendix by Don B. Zagier
title_full_unstemmed	Graphs on surfaces and their applications Sergei K. Lando ; Alexander K. Zvonkin. Appendix by Don B. Zagier
title_short	Graphs on surfaces and their applications
title_sort	graphs on surfaces and their applications
topic	Topologische Graphentheorie (DE-588)4341226-9 gnd
topic_facet	Topologische Graphentheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010211596&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV014813608 (DE-604)BV024126459
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