Vassiliev knot invariants. III: Forest algebra and weighted graphs

Abstract

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.

A weighted graph polynomial from chromatic invariants of knots

“…In a sense this is not too surprising in view of the established relation between Tutte polynomials and knot polynomials, see for example Jaeger [8] or Jaeger, Vertigan and Welsh [9]. Here we define a polynomial of weighted graphs which contains the invariants of [4] as a specialisation but moreover has a wide range of other specialisations in combinatorics. These include the Tutte polynomial, stability polynomial and matching polynomial of ordinary graphs and a polymatroid polynomial studied by Oxiey and Whittle [16].…”

“…In a series of three papers [2], [3], [4], Chmutov, Duzhin and Lando examine the combinatorial aspects of the relationship between chord diagrams and Vassiliev invariants of knots.…”

“…A key concept in the treatment in [4] is the homomorphism associating a chord diagram with its intersection graph, and this turns out to be just a map on weighted graphs modulo a weighted chromatic relation. This is essentially the same as the study of a chromatic type polynomial on weighted graphs.…”

“…This is essentially the same as the study of a chromatic type polynomial on weighted graphs. However, as the authors point out, the treatment given in [4] does not allow an extension to more general Tutte Grothendieck invariants. This paper originated in an attempt to clarify the reasons for this and we show that a fairly simple modification of the definition of [4] does allow such an extension.…”

“…However, as the authors point out, the treatment given in [4] does not allow an extension to more general Tutte Grothendieck invariants. This paper originated in an attempt to clarify the reasons for this and we show that a fairly simple modification of the definition of [4] does allow such an extension. In a sense this is not too surprising in view of the established relation between Tutte polynomials and knot polynomials, see for example Jaeger [8] or Jaeger, Vertigan and Welsh [9].…”