Skip to main content
×
Home
    • Aa
    • Aa

On the computational complexity of the Jones and Tutte polynomials

  • F. Jaeger (a1), D. L. Vertigan (a2) and D. J. A. Welsh (a2)
Abstract
Abstract

We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]L. Berman and J. Hartmanis . On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6 (1977), 305321.

[3]J. A. Bondy and U. S. R. Murty . Graph Theory with Applications (American Elsevier and Macmillan, 1976).

[4]T. H. Brylawski . A decomposition for combinatorial geometries. Trans. Amer. Math. Soc. 171 (1972), 235282.

[8]H. Crapo . The Tutte polynomial. Aequationes Math. 3 (1969), 211229.

[9]M. E. Fisher . On the dimer solution of planar Ising models. J. Math. Phys. 7 (1966), 17761781.

[11]M. R. Garey , D. S. Johnson and L. Stockmeyer . Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1 (1976), 237267.

[13]M. Grötschel , L. Lovász and A. Schrijver . Geometric Algorithms and Combinatorial Optimization (Springer-Verlag, 1988).

[14]F. Jaeger . On Tutte polynomials and cycles of plane graphs. J. Combin. Theory Ser. B 44 (1988), 129146.

[15]F. Jaeger . On Tutte polynomials and link polynomials. Proc. Amer. Math. Soc. 103 (1988), 647654.

[16]F. Jaeger . On Tutte polynomials and bicycle dimension of ternary matroids. Proc. Amer. Math. Soc. 107 (1989), 1725.

[19]M. R. Jerrum . 2-dimensional monomer-dimer systems are computationally intractable. J. Statist. Phys. 48 (1987), 121134.

[21]V. F. R. Jones . A polynomial invariant for knots via Von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103111.

[24]M. Las Vergnas . Convexity in oriented matroids. J. Combin. Theory Ser. B 29 (1980), 231243.

[26]W. B. R. Lickorish . Polynomials for links. Bull. London Math. Soc. 20 (1988), 558588.

[27]N. Linial . Hard enumeration problems in geometry and combinatorics. SIAM J. Algebraic Discrete Methods 7 (1986), 331335.

[30]P. Martin . Remarkable valuation of the dichromatic polynomial of planar multigraphs. J. Combin. Theory Ser. B 24 (1978), 318324.

[32]J. S. Provan . The complexity of reliability computations in planar and acyclic graphs. SIAM J. Comput. 15 (1986), 694702.

[33]J. S. Provan and M. O. Ball . The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12 (1983), 777788.

[35]P. Rosenstiehl and R. C. Read . On the principal edge tripartition of a graph. Ann. Discrete Math. 3 (1978), 195226.

[36]P. D. Seymour . Decomposition of regular matroids. J. Combin. Theory Ser. B 28 (1980), 305359.

[37]P. D. Seymour . Nowhere-zero 6-flows. J. Combin. Theory Ser. B 30 (1981), 130135.

[38]R. P. Stanley . Enumerative Combinatorics, vol. 1 (Wadsworth & Brooks/Cole, 1986).

[39]M. B. Thistlethwaite . A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297309.

[40]M. B. Thistlethwaite . On the Kauffman polynomial of an adequate link. Invent. Math. 93 (1988), 285296.

[42]L. G. Valiant . The complexity of computing the permanent. Theoret. Comput. Sci. 8 (1979), 189201.

[43]L. G. Valiant . The complexity of enumeration and reliability problems. SIAM J. Comput. 8 (1979), 410421.

[47]H. Whitney . A logical expansion in mathematics. Bull. Amer. Math. Soc. 38 (1932), 572579.

[48]H. Whitney . On the abstract properties of linear dependence. Amer. J. Math. 57 (1935), 509533.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×