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Tutte polynomials and bicycle dimension of ternary matroids. (English) Zbl 0669.05023
Let \(M\) be a ternary matroid, \(t(M,x,y)\) be its Tutte polynomial and \(d(M)\) be the dimension of the bicycle space of any representation of \(M\) over \(GF(3)\).
We show that, for \(j=e^{2i\pi /3}\), the modulus of the complex number \(t(M,j,j^ 2)\) is equal to (\(\sqrt{3})^{d(M)}\). The proof relies on the study of the weight enumerator \(W_{{\mathcal C}}(y)\) of the cycle space \({\mathcal C}\) of a representation of \(M\) over \(GF(3)\) evaluated at \(y=j\). The main tool is the concept of principal quadripartition of \({\mathcal C}\) which allows a precise analysis of the evolution of the relevant invariants under deletion and contraction of elements.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
05A15 Exact enumeration problems, generating functions
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[1] Thomas H. Brylawski, A decomposition for combinatorial geometries, Trans. Amer. Math. Soc. 171 (1972), 235 – 282. · Zbl 0224.05007
[2] T. H. Brylawski and D. Lucas, Uniquely representable combinatorial geometries, Proc. of Int. Colloq. in Combinatorial Theory, Rome, Italy, 1973; Atti Dei Convegni Lincei 17, Tomo 1 (1976), 83-104. · Zbl 0392.51007
[3] Henry H. Crapo, The Tutte polynomial, Aequationes Math. 3 (1969), 211 – 229. · Zbl 0197.50202 · doi:10.1007/BF01817442 · doi.org
[4] Curtis Greene, Weight enumeration and the geometry of linear codes, Studies in Appl. Math. 55 (1976), no. 2, 119 – 128. · Zbl 0331.05019
[5] Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195 – 242. · Zbl 0657.57001 · doi:10.2307/2323625 · doi.org
[6] W. B. R. Lickorish and K. C. Millett, Some evaluations of link polynomials, Comment. Math. Helv. 61 (1986), no. 3, 349 – 359. · Zbl 0607.57003 · doi:10.1007/BF02621920 · doi.org
[7] M. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, New York, Oxford, 1978. · Zbl 0369.94008
[8] P. Rosenstiehl and R. C. Read, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978), 195 – 226. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). · Zbl 0392.05059
[9] Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297 – 309. · Zbl 0622.57003 · doi:10.1016/0040-9383(87)90003-6 · doi.org
[10] W. T. Tutte, A ring in graph theory, Proc. Cambridge Philos. Soc. 43 (1947), 26 – 40. · Zbl 0031.41803
[11] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canadian J. Math. 6 (1954), 80 – 91. · Zbl 0055.17101
[12] D. J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. · Zbl 0343.05002
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