Colored Tutte Polynomials and Kauuman Brackets for Graphs of Bounded Tree Width

Jones polynomials and Kauuman 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 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 deened as the tree width of its graphical presentation 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 Kauuman square bracket L] introduced by Bollobas and Riordan can be computed in polynomial time. In particular, the classical Kauuman bracket hLi and the Jones polynomial of links of tree width at most k are computable in polynomial time.