Algebraic & Geometric Topology

Partial duals of plane graphs, separability and the graphs of knots

Iain Moffatt

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There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.

Article information

Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1099-1136.

Received: 10 January 2012
Revised: 23 February 2012
Accepted: 25 February 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 57M15: Relations with graph theory [See also 05Cxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 05C75: Structural characterization of families of graphs

$1$–sum checkerboard graph dual embedded graph knots and links Partial duality plane graph ribbon graph separability Tait graph Turaev surface


Moffatt, Iain. Partial duals of plane graphs, separability and the graphs of knots. Algebr. Geom. Topol. 12 (2012), no. 2, 1099--1136. doi:10.2140/agt.2012.12.1099.

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  • T Abe, The Turaev genus of an adequate knot, Topology Appl. 156 (2009) 2704–2712
  • A Champanerkar, I Kofman, N Stoltzfus, Graphs on surfaces and Khovanov homology, Algebr. Geom. Topol. 7 (2007) 1531–1540
  • S Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory Ser. B 99 (2009) 617–638
  • S Chmutov, I Pak, The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial, Mosc. Math. J. 7 (2007) 409–418, 573
  • S Chmutov, J Voltz, Thistlethwaite's theorem for virtual links, J. Knot Theory Ramifications 17 (2008) 1189–1198
  • O T Dasbach, D Futer, E Kalfagianni, X-S Lin, N W Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2008) 384–399
  • O T Dasbach, D Futer, E Kalfagianni, X-S Lin, N W Stoltzfus, Alternating sum formulae for the determinant and other link invariants, J. Knot Theory Ramifications 19 (2010) 765–782
  • O T Dasbach, A M Lowrance, Turaev genus, knot signature, and the knot homology concordance invariants, Proc. Amer. Math. Soc. 139 (2011) 2631–2645
  • J A Ellis-Monaghan, I Moffatt, Twisted duality for embedded graphs, Trans. Amer. Math. Soc. 364 (2012) 1529–1569
  • D Futer, E Kalfagianni, J S Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008) 429–464
  • D Futer, E Kalfagianni, J S Purcell, Symmetric links and Conway sums: volume and Jones polynomial, Math. Res. Lett. 16 (2009) 233–253
  • S Huggett, I Moffatt, Bipartite partial duals and circuits in medial graphs, to appear in Combinatorica
  • S Huggett, I Moffatt, N Virdee, On the graphs of link diagrams and their parallels, to appear in Math. Proc. Cambridge Philos. Soc.
  • T Krajewski, V Rivasseau, F Vignes-Tourneret, Topological graph polynomial and quantum field theory Part II: Mehler kernel theories, Ann. Henri Poincaré 12 (2011) 483–545
  • A M Lowrance, On knot Floer width and Turaev genus, Algebr. Geom. Topol. 8 (2008) 1141–1162
  • I Moffatt, Separability and the genus of a partial dual
  • I Moffatt, Partial duality and Bollobás and Riordan's ribbon graph polynomial, Discrete Math. 310 (2010) 174–183
  • I Moffatt, A characterization of partially dual graphs, J. Graph Theory 67 (2011) 198–217
  • I Moffatt, Unsigned state models for the Jones polynomial, Ann. Comb. 15 (2011) 127–146
  • V G Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, Enseign. Math. 33 (1987) 203–225
  • F Vignes-Tourneret, The multivariate signed Bollobás–Riordan polynomial, Discrete Math. 309 (2009) 5968–5981
  • F Vignes-Tourneret, Non-orientable quasi-trees for the Bollobás–Riordan polynomial, European J. Combin. 32 (2011) 510–532
  • T Widmer, Quasi-alternating Montesinos links, J. Knot Theory Ramifications 18 (2009) 1459–1469