We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás-Riordan polynomials.

With hundreds of worked examples, exercises and illustrations, this detailed exposition of the theory of Vassiliev knot invariants opens the field to students with little or no knowledge in this area. It also serves as a guide to more advanced material. The book begins with a basic and informal introduction to knot theory, giving many examples of knot invariants before the class of Vassiliev invariants is introduced. This is followed by a detailed study of the algebras of Jacobi diagrams and 3-graphs, and the construction of functions on these algebras via Lie algebras. The authors then describe two constructions of a universal invariant with values in the algebra of Jacobi diagrams: via iterated integrals and via the Drinfeld associator, and extend the theory to framed knots. Various other topics are then discussed, such as Gauss diagram formulae, before the book ends with Vassiliev's original construction.

We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás-Riordan polynomial RG L of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of planar graphs.

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The main tool used in the investigation of Vassiliev knot invariants is the Hopf algebra of chord diagrams CDL1]. This algebra, simple as it seems at rst sight, upon a closer examination proves to be a rather complicated object. It is su cient to say that, up to now, the number of its primitive generators is known only in degrees no greater than 9. A standard way to tackle a complex mathematical object O is through the study of its subobjects and its quotient objects. An ideal situation is when you can distinguish a simple subobject S O such that the corresponding quotient object O=S is also simple enough and so that the properties of the entire O are completely determined by the properties of S and O=S. With the Hopf algebra of chord diagrams, we were not able to achieve this goal. The best we could do was to nd a simple (but non-trivial) subalgebra A M and a simple quotient algebra W de ned by an epimorphism M ! W. These two algebras, A and W, are, however, not complementary to each other. Quite on the contrary, the composition map in the short non-exact sequence 0 ! A ! M ! W ! 0 proves to be an isomorphism between A and W.