Colored Tutte polynomials and Kaufman brackets for graphs of bounded tree width

  • Authors:

  • J. A. Makowsky

  • Affiliations:

  • Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel

  • Venue:
  • SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2001

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Abstract

Jones polynomials and Kauffman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials by a result of Thistlethwaite (1988), but in general one needs Kauffman's Tutte polynomials for signed graphs (1989), further generalized to colored Tutte polynomials, as introduced by Bollobas and Riordan (1999). Knots and links can be presented as labeled planar graphs. The tree width of a link L is defined as the minimal tree width of its graphical presentations D(L) as crossing diagrams. We show that the colored Tutte polynomial can be computed in polynomial time for graphs of tree width at most k. Hence, for (not necessarily alternating) knots and links of tree width at most k, even the Kauffman square bracket [L] introduced by Bollobas and Riordan can be computed in polynomial time. In particular, the classical Kauffman bracket (L) and the Jones polynomial of links of tree width at most k are computable in polynomial time.