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Graphs on Surfaces and Their Applications

0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.-…
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Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only…
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An algebro-geometric proof of Witten's conjecture

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating…
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Lectures on Generating Functions

Formal power series and generating functions. Operations with formal power series. Elementary generating functions Generating functions for well-known sequences Unambiguous formal grammars. The…
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Plane and Projective Meanders

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On Hurwitz numbers and Hodge integrals

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Mutant knots and intersection graphs

We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram…
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J-invariants of plane curves and framed chord diagrams

Arnold defined J-invariants of general plane curves as functions on classes of such curves that jump in a prescribed way when passing through curves with self-tangency. The coalgebra of framed chord…
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On a Hopf Algebra in Graph Theory

    S. Lando
    Mathematics
    J. Comb. Theory B
  • 1 September 2000
The study is introduced of a bialgebra of graphs, which is similar to the ring of graphs introduced by W. T. Tutte in 1946, and of the dual bial algebra of 4-invariants, which determines Vassiliev invariants of knots.
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Weight systems and invariants of graphs and embedded graphs

Certain recent papers suggest an approach to extending weight systems and graph invariants to arbitrary embedded graphs, which is based on the study of the corresponding Hopf algebra structures; this approach is described.
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