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Generalised Dualities

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Graphs on Surfaces

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Abstract

This chapter introduces the framework upon which we will build the rest of this monograph. We begin by considering geometric duals and Petrials, observing that Petriality and geometric duality result from local operations on each edge of an embedded graph. These local operations applied to subsets of the edge set result in partial Petrality and partial duality. We provide constructions for partial duals and partial Petrials in various realisations of embedded graphs.The two operations of partial Petrality and partial duality give rise to an action of the symmetric group on embedded graphs with a distinguished set of edges. This group action leads to twisted duality, which assimilates several types of duality from the literature, including geometric duality, direct derivatives, Petrie duals, and partial duality. We concluded by defining the ribbon group and describing how twisted duality can be obtained as orbits under the ribbon group action on the set of edge-ordered embedded graphs.

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References

  1. Abe T (2009) The Turaev genus of an adequate knot. Topology Appl 156:2704–2712

    Article  MathSciNet  MATH  Google Scholar 

  2. Aigner M (1997) The Penrose polynomial of a plane graph. Math Ann 307:173–189

    Article  MathSciNet  MATH  Google Scholar 

  3. Askanazi R, Chmutov S, Estill C, Michel J, Stollenwerk P (2013) Polynomial invariants of graphs on surfaces. Quantum Topol 3:77–90

    Article  MathSciNet  Google Scholar 

  4. Bollobás B (1998) Modern graph theory. In: Graduate texts in mathematics, vol 184. Springer, New York

    Google Scholar 

  5. Bollobás B, Riordan O (2001) A polynomial for graphs on orientable surfaces. Proc London Math Soc 83:513–531

    Article  MathSciNet  MATH  Google Scholar 

  6. Bollobás B, Riordan O (2002) A polynomial of graphs on surfaces. Math Ann 323:81–96

    Article  MathSciNet  MATH  Google Scholar 

  7. Bondy J, Murty U (2008) Graph theory. In: Graduate texts in mathematics, vol 244. Springer, New York

    Google Scholar 

  8. Bradford R, Butler C, Chmutov S (2012) Arrow ribbon graphs. J Knot Theory Ramifications 21:1240002

    Article  MathSciNet  Google Scholar 

  9. Burde G, Zieschang H (2003) Knots, 2nd edn. de Gruyter studies in mathematics, vol 5. Walter de Gruyter, Berlin

    Google Scholar 

  10. Brylawski T (1982) The Tutte polynomial. Part 1: General theory. In: Barlotti A (ed) Matroid theory and its applications, proceedings of the third international mathematical summer center (C.I.M.E. 1980), pp 125–275

    Google Scholar 

  11. Brylawski T, Oxley J (1992) The Tutte polynomial and its applications. In: White N (ed) Matroid applications, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge

    Google Scholar 

  12. Carter J, Kamada S, Saito M (2002) Stable equivalences of knots on surfaces and virtual knot cobordisms. J Knot Theory Ramifications 11:311–322

    Article  MathSciNet  MATH  Google Scholar 

  13. Champanerkar A, Kofman I (2009) Spanning trees and Khovanov homology. Proc Amer Math Soc 137:2157–2167

    Article  MathSciNet  MATH  Google Scholar 

  14. Champanerkar A, Kofman I, Stoltzfus N (2007) Graphs on surfaces and Khovanov homology. Algebr Geom Topol 7:1531–1540

    Article  MathSciNet  MATH  Google Scholar 

  15. Champanerkar A, Kofman I, Stoltzfus N (2011) Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial. Bull Lond Math Soc 43:972–984

    Article  MathSciNet  MATH  Google Scholar 

  16. Chmutov S (2009) Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial. J Combin Theory Ser B 99:617–638

    Article  MathSciNet  MATH  Google Scholar 

  17. Chmutov S, Pak I (2007) The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial. Mosc Math J 7:409–418

    MathSciNet  MATH  Google Scholar 

  18. Chmutov S, Voltz J (2008) Thistlethwaite’s theorem for virtual links. J Knot Theory Ramifications 17:1189–1198

    Article  MathSciNet  MATH  Google Scholar 

  19. Crowell R, Fox R (1977) Introduction to knot theory. In: Graduate texts in mathematics, vol 57. Springer, New York

    Google Scholar 

  20. Cotta-Ramusino P, Rinaldi M (1991) On the algebraic structure of link-diagrams on a 2-dimensional surface. Comm Math Phys 138:137–173

    Article  MathSciNet  MATH  Google Scholar 

  21. Dasbach O, Futer D, Kalfagianni E, Lin X-S, Stoltzfus N (2008) The Jones polynomial and graphs on surfaces. J Combin Theory Ser B 98:384–399

    Article  MathSciNet  MATH  Google Scholar 

  22. Dasbach O, Futer D, Kalfagianni E, Lin X-S, Stoltzfus N (2010) Alternating sum formulae for the determinant and other link invariants. J Knot Theory Ramifications 19:765–782

    Article  MathSciNet  MATH  Google Scholar 

  23. Dasbach O, Lowrance A (2011) Turaev genus, knot signature, and the knot homology concordance invariants. Proc Amer Math Soc 139:2631–2645

    Article  MathSciNet  MATH  Google Scholar 

  24. Diestel R (2010) Graph theory. In: Graduate texts in mathematics, vol 173, 4th edn. Springer, Heidelberg

    Google Scholar 

  25. Dye H, Kauffman L (2009) Virtual crossing number and the arrow polynomial. J Knot Theory Ramifications 18:1335–1357

    Article  MathSciNet  MATH  Google Scholar 

  26. Ellis-Monaghan J, Merino C (2011) Graph polynomials and their applications I: the Tutte polynomial. In: Dehmer M (ed) Structural analysis of complex networks. Birkhauser, New York, pp 219–255

    Chapter  Google Scholar 

  27. Ellis-Monaghan J, Merino C (2011) Graph polynomials and their applications II: interrelations and interpretations. In: Dehmer M (ed) Structural analysis of complex networks. Birkhauser, New York, pp 257–292

    Chapter  Google Scholar 

  28. Ellis-Monaghan J, Moffatt I (2013) A Penrose polynomial for embedded graphs. Eur J Combin 34:424–445

    Article  MathSciNet  MATH  Google Scholar 

  29. Ellis-Monaghan J, Moffatt I. Evaluations of topological Tutte polynomials. arXiv:1108.3321 (preprint)

    Google Scholar 

  30. Ellis-Monaghan J, Moffatt I (2012) Twisted duality and polynomials of embedded graphs. Trans Amer Math Soc 364:1529–1569

    Article  MathSciNet  MATH  Google Scholar 

  31. Ellis-Monaghan J, Sarmiento I (2001) Medial graphs and the Penrose polynomial. Congr Numer, vol 150, pp 211–222

    MathSciNet  MATH  Google Scholar 

  32. Ellis-Monaghan J, Sarmiento I (2002) Generalized transition polynomials. Congr Numer 155:57–69

    MathSciNet  MATH  Google Scholar 

  33. Ellis-Monaghan J, Sarmiento I. A duality relation for the topological Tutte polynomial, talk at the AMS eastern section meeting special session on graph and matroid invariants, bard college. http://academics.smcvt.edu/jellis-monaghan/\#Research. Accessed 10 Sept 2005

  34. Ellis-Monaghan J, Sarmiento I (2011) A recipe theorem for the topological Tutte polynomial of Bollobás and Riordan. Eur J Combin 32:782–794

    Article  MathSciNet  MATH  Google Scholar 

  35. Fox R, Artin E (1948) Some wild cells and spheres in three-dimensional space. Ann Math 49:979–990

    Article  MathSciNet  MATH  Google Scholar 

  36. Fox R (1949) A remarkable simple closed curve. Ann Math 50:264–265

    Article  MATH  Google Scholar 

  37. Freyd P, Hoste J, Lickorish W,  Millett K,  Ocneanu A,  Yetter D (1985) A new polynomial invariant of knots and links. Bull Amer Math Soc (N.S.) 12:239–246

    Google Scholar 

  38. Futer D, Kalfagianni E, Purcell J (2008) Dehn filling, volume, and the Jones polynomial. J Differ Geom 78:429–464

    MathSciNet  MATH  Google Scholar 

  39. Futer D, Kalfagianni E, Purcell J (2009) Symmetric links and Conway sums: volume and Jones polynomial. Math Res Lett 16:233–253

    MathSciNet  MATH  Google Scholar 

  40. Goussarov M, Polyak M, Viro O (2000) Finite-type invariants of classical and virtual knots. Topology 39:1045–1068

    Article  MathSciNet  MATH  Google Scholar 

  41. Gross J, Tucker T (1987) Topological graph theory, Wiley-Interscience, New York

    MATH  Google Scholar 

  42. Gurau R (2010) Topological graph polynomials in colored group field theory. Ann Henri Poincaré 11:565–584

    Article  MathSciNet  MATH  Google Scholar 

  43. Huggett S, Moffatt I (2011) Expansions for the Bollobás-Riordan and Tutte polynomials of separable ribbon graphs. Ann Comb 15:675–706

    Article  MathSciNet  MATH  Google Scholar 

  44. Huggett S, Moffatt I. Bipartite partial duals and circuits in medial graphs, Combinatorica (in press) arXiv:1106.4189

    Google Scholar 

  45. Huggett S, Moffatt I, Virdee N (2012) On the Seifert graphs of link diagrams and their parallels. Math Proc Cambridge Philos Soc 153:123–145

    Article  MathSciNet  MATH  Google Scholar 

  46. Hoste J, Przytycki J (1989) An invariant of dichromatic links. Proc Amer Math Soc 105:1003–1007

    Article  MathSciNet  MATH  Google Scholar 

  47. Inoue K, Kaneto T (1994) A Jones type invariant of links in the product space of a surface and the real line. J Knot Theory Ramifications 3:153–161

    Article  MathSciNet  MATH  Google Scholar 

  48. Jaeger F (1988) Tutte polynomials and link polynomials. Proc Amer Math Soc 103:647–654

    Article  MathSciNet  MATH  Google Scholar 

  49. Jaeger F (1988) On Tutte polynomials and cycles of plane graphs. J Combin Theory Ser B 44:127–146

    Article  MathSciNet  MATH  Google Scholar 

  50. Jaeger F (1990) On transition polynomials of 4-regular graphs. In: Hahn G. et al (eds) Cycles and rays (Montreal, PQ, 1987). NATO advanced science institutes series c: mathematical and physical sciences, vol 301. Kluwer Academic, Dordrecht, pp 123–150

    Chapter  Google Scholar 

  51. Jin X, Zhang F (2012) The Homfly and dichromatic polynomials. Proc Amer Math Soc 140:1459–1472

    Article  MathSciNet  MATH  Google Scholar 

  52. Jones V (1985) A polynomial invariant for knots via von Neumann algebras. Bull Amer Math Soc 12:103–111

    Article  MathSciNet  MATH  Google Scholar 

  53. Kauffman L (1987) State models and the Jones polynomial. Topology 26:395–407

    Article  MathSciNet  MATH  Google Scholar 

  54. Kauffman L (1990) An invariant of regular isotopy. Trans Amer Math Soc 312:417–471

    Article  MathSciNet  Google Scholar 

  55. Kauffman L (1999) Virtual knot theory. Eur J Combin 20:663–690

    Article  MathSciNet  MATH  Google Scholar 

  56. Kauffman L (2000) A survey of virtual knot theory. In: Knots in Hellas ’98 (Delphi), Ser. Knots Everything, vol 24. World Science, River Edge, pp 143–202

    Google Scholar 

  57. Kauffman L (2012) Introduction to virtual knot theory. J. Knot Theory Ramifications 21:1240007

    Article  MathSciNet  Google Scholar 

  58. Kauffman L, Manturov V (2006) Virtual knots and links, Tr. Mat. Inst. Steklova Geom. Topol., Diskret. Geom. i Teor. Mnozh., translation in Proc. Steklov Inst. Math. 25:114–133; 252:104–121.

    Google Scholar 

  59. Kamada N (2002) On the Jones polynomials of checkerboard colorable virtual links. Osaka J Math 39:325–333

    MathSciNet  MATH  Google Scholar 

  60. Kamada N (2004) Span of the Jones polynomial of an alternating virtual link. Algebr Geom Topol 4:1083–1101

    Article  MathSciNet  MATH  Google Scholar 

  61. Korn M, Pak I. Combinatorial evaluations of the Tutte polynomial (preprint)

    Google Scholar 

  62. Kotzig A (1968) Eulerian lines in finite 4-valent graphs and their transformations. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic, New York, pp 219–230

    Google Scholar 

  63. Krajewski T, Rivasseau V, Vignes-Tourneret F (2011) Topological graph polynomials and quantum field theory. Part II: Mehler kernel theories. Ann Henri Poincaré 12:1–63

    Article  MathSciNet  Google Scholar 

  64. Lando S, Zvonkin A (2004) Graphs on surfaces and their applications, encyclopaedia of mathematical sciences, low-dimensional topology, II. vol 141. Springer, Berlin

    Book  Google Scholar 

  65. Las Vergnas M (1981) Eulerian circuits of 4-valent graphs imbedded in surfaces. In: Algebraic methods in graph theory, vols I, II (Szeged, 1978), vol 25. Colloquia Mathematica Societatis Jnos Bolyai, North-Holland, Amsterdam, New York, pp 451–477

    Google Scholar 

  66. Las Vergnas M (1978) Sur les activités des orientations d’une géométrie combinatoire. In: colloque mathématiques discrètes: codes et hypergraphes (Brussels, 1978). cahiers centre Études rech Opér 20:293–300

    Google Scholar 

  67. Las Vergnas M (1980) On the Tutte polynomial of a morphism of matroids. Ann Discrete Math 8:7–20

    Article  MathSciNet  MATH  Google Scholar 

  68. Lickorish W 1997 An introduction to knot theory. In: Graduate texts in mathematics, vol 175. Springer, New York

    Google Scholar 

  69. Lieberum J (2002) Skein modules of links in cylinders over surfaces. Int J Math Math Sci 32:515–554

    Article  MathSciNet  MATH  Google Scholar 

  70. Loebl M, Moffatt I (2008) The chromatic polynomial of fatgraphs and its categorification. Adv Math 217:1558–1587

    Article  MathSciNet  MATH  Google Scholar 

  71. Lowrance A (2008) On knot Floer width and Turaev genus. Algebr Geom Topol 8:1141–1162

    Article  MathSciNet  MATH  Google Scholar 

  72. Manturov V (2004) Knot theory. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  73. Massey W (1977) Algebraic topology: an introduction. In: Graduate texts in mathematics, vol 56. Springer, New York

    Google Scholar 

  74. Milnor J (1964) Most knots are wild. Fund Math 54:335–338

    MathSciNet  MATH  Google Scholar 

  75. Moffatt I (2008) Knot invariants and the Bollobás-Riordan polynomial of embedded graphs. Eur J Combin 29: 95–107

    Article  MathSciNet  MATH  Google Scholar 

  76. Moffatt I (2010) Partial duality and Bollobás and Riordan’s ribbon graph polynomial. Discrete Math 310:174–183

    Article  MathSciNet  MATH  Google Scholar 

  77. Moffatt I (2011) Unsigned state models for the Jones polynomial. Ann Comb 15: 127–146

    Article  MathSciNet  MATH  Google Scholar 

  78. Moffatt I (2011) A characterization of partially dual graphs. J Graph Theory 67:198–217

    Article  MathSciNet  MATH  Google Scholar 

  79. Moffatt I (2012) Partial duals of plane graphs, separability and the graphs of knots. Algebr Geom Topol 12:1099–1136

    Article  MathSciNet  MATH  Google Scholar 

  80. Moffatt I (2013) Separability and the genus of a partial dual. Eur J Combin 34:355–378

    Article  MathSciNet  MATH  Google Scholar 

  81. Mohar B, Thomassen C (2001) Graphs on surfaces (Johns Hopkins studies in the mathematical sciences). John Hopkins University Press, Baltimore

    Google Scholar 

  82. Nelson S (2001) Unknotting virtual knots with Gauss diagram forbidden moves. J Knot Theory Ramifications 10:931–935

    Article  MathSciNet  MATH  Google Scholar 

  83. Oxley J, Welsh D (1979) The Tutte polynomial and percolation. In: Bondy JA, Murty USR (eds) Graph theory and related topics. Academic, London

    Google Scholar 

  84. Penrose R (1971) Applications of negative dimensional tensors, combinatorial mathematics and its applications. In: Proceedings of the conference held in Oxford in 1969. Academic, London, pp 221–244

    Google Scholar 

  85. Polyak M (2010) Minimal generating sets of Reidemeister moves. Quantum Topol 1:399–411

    Article  MathSciNet  MATH  Google Scholar 

  86. Przytycki J (1999) Fundamentals of Kauffman bracket skein modules. Kobe J Math 16:45–66

    MathSciNet  MATH  Google Scholar 

  87. Przytycki J, Traczyk P (1988) Invariants of links of Conway type. Kobe J Math 4:115–139

    MathSciNet  Google Scholar 

  88. Rolfsen D (1976) Knots and links. Mathematics lecture series, No. 7. Publish or Perish, Berkeley, California

    Google Scholar 

  89. Tanasa A (2011) Generalization of the Bollobás-Riordan polynomial for tensor graphs. J Math Phys 52:073514

    Article  MathSciNet  Google Scholar 

  90. Tanasa A (2012) Some combinatorial aspects of quantum field theory. Seminaire Lotharingien de Combinatoire, B65g

    Google Scholar 

  91. Tanasa A (2012) Multi-orientable group field theory. J Phys A: Math Theor 45:165401

    Article  MathSciNet  Google Scholar 

  92. Thistlethwaite M (1987) A spanning tree expansion of the Jones polynomial. Topology 26:297–309

    Article  MathSciNet  MATH  Google Scholar 

  93. Traldi L (1989) A dichromatic polynomial for weighted graphs and link polynomials. Proc Amer Math Soc 106:279–286

    Article  MathSciNet  MATH  Google Scholar 

  94. Turaev V (1990) The Conway and Kauffman modules of the solid torus. J Soviet Math 52:2799–2805

    Article  MathSciNet  MATH  Google Scholar 

  95. Turaev V (1987) A simple proof of the Murasugi and Kauffman theorems on alternating links. Enseign Math 33:203–225

    MathSciNet  MATH  Google Scholar 

  96. Vignes-Tourneret F (2009) The multivariate signed Bollobás-Riordan polynomial. Discrete Math 309:5968–5981

    Article  MathSciNet  MATH  Google Scholar 

  97. Vignes-Tourneret F (2011) Non-orientable quasi-trees for the Bollobás-Riordan polynomial. Eur J Combin 32: 510–532

    Article  MathSciNet  MATH  Google Scholar 

  98. West D (1996) Introduction to graph theory (2nd edn). Prentice Hall, Upper Saddle River

    Google Scholar 

  99. Widmer T (2009) Quasi-alternating Montesinos links. J Knot Theory Ramifications 18: 1459–1469

    Article  MathSciNet  MATH  Google Scholar 

  100. Wilson S (1979) Operators over regular maps. Pacific J Math 81:559–568

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Saint Michael’s College, Colchester, VT, 05439, USA

    Joanna A. Ellis-Monaghan

  2. Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK

    Iain Moffatt

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  1. Joanna A. Ellis-Monaghan
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  2. Iain Moffatt
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© 2013 Joanna A. Ellis-Monaghan, Iain Moffatt

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Ellis-Monaghan, J.A., Moffatt, I. (2013). Generalised Dualities. In: Graphs on Surfaces. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6971-1_2

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