Annals of Combinatorics 4 (2000) 227-236


The Polychromate and a Chord Diagram

Irasema Sarmiento

Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 3, D-14195 Berlin, Germany
sarmient@math.fu-berlin.de

Received June 4, 1998

AMS Subject Classification: 05C99

Abstract. In [11] Noble and Welsh defined the weighted polynomial of a graph. It permits the extension of the work of Chmutov, Duzhin and Lando [8] to more general Tutte Grothendieck invariants. The U-polynomial is a specialization of the weighted polynomial when all the vertices have weight one. It corresponds to the original motivating situation of intersection graphs of chord diagrams of knots. We show that the polychromate and the U-polynomial determine one another.

Keywords: Chord diagram, knots, Vassiliev invariants, polychromate, embedding, matroid, geometry, graph


References

1.  T.H. Brylawski, A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171 (1972) 235–282.

2.  T.H. Brylawski, Intersection theory for graphs, J. Combin. Theory Ser. B 30 (1981) 233–246.

3.  T.H. Brylawski, Intersection theory for embeddings of matroids into uniform geometries, Studies in Appl. Math. 61 (1979) 211–244.

4.  T.H. Brylawski, The Tutte polynomial, Part I: General theory, In: Matroid Theory and its Applications, A. Barlotti, Ed., C.I.M.E., Liguori, Naples, 1980, pp. 125–175.

5.  T.H. Brylawski and J. Oxley, The Tutte polynomial and its applications, In: Matroid Applications, N. White, Ed., Cambridge University Press, Cambridge, 1992, pp. 123–225.

6.  S.V. Chmutov, S.V. Duzhin, and S.K. Lando, Vassiliev knot invariants: I. Introduction, Adv. Sov. Math. 21 (1994) 117–126.

7.  S.V. Chmutov, S.V. Duzhin, and S.K. Lando, Vassiliev knot invariants: II. Intersection graph for trees, Adv. Sov. Math. 21 (1994) 127–134.

8.  S.V. Chmutov, S.V. Duzhin, and S.K. Lando, Vassiliev knot invariants: III. Forest algebra and weighted graphs, Adv. Sov. Math. 21 (1994) 135–145.

9.  M. Kontsevich, Vassiliev’s knot invariants, Adv. Sov. Math. 16 (1993) 137–150.

10.  I.G. MacDonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford University Press, New York, 1979.

11.  S.D. Noble and D.J.A. Welsh, A weighted graph polynomial from chromatic invariants of knots, Annales de l’Institute Fourier 49 (1999) 101–131.

12.  J.G. Oxley, Matroid Theory, Oxford University Press, Oxford, 1992.

13.  R.P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995) 166–194.

14.  R.P. Stanley, Graph colourings and related symmetric functions: Ideas and applications, Discrete Math., to appear.

15.  V.A. Vassiliev, Cohomology of knot spaces, Adv. Sov. Math. 1 (1990) 9–21.

16.  S. Willerton, Review of Chmutov, Duzhin, and Lando, Vassiliev knot invariants I-III, Mathematical Reviews 96i57002, 1996.


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