The parametrized complexity of knot polynomials

We study the parametrized complexity of the knot (and link) polynomials known as Jones polynomials, Kauuman polynomials and Hommy-polynomials. It is known that computing these polynomials is ]P hard in general. We look for parameters of the combinatorial presentation of knots and links which make the computation of these polynomials to be xed parameter tractable, i.e. to be in FPT. If the link is explicitly presented as a closed braid, the number of its strands is known to be such a parameter. In a generalization thereof, if the link is explicitly presented as a combination of compositions and rotations of k-tangles the link is called k-algebraic, and its algebraicity k is such a parameter. The previously known proofs that for this parameter the link polynomials are in FPT uses the so called skein-modules, and is algebraic in its nature. Furthermore, it is not clear how to nd such an algebraic presentation from a given link diagram. We look at the treewidth of two combinatorial presentation of links: the crossing diagram and its shading, a signed graph. We show that the treewidth of these two presentations and the algebraicity of links are all linearly related to each other. Furthermore, we characterize the k-algebraic links using the pathwidth of the crossing diagram. Using this, we can apply algorithms for testing xed treewidth to nd k-algebraic presentations in polynomial time. From this we can conclude that also treewidth and pathwidth are parameters of link diagrams for which the knot polynomials are FPT. For the Kauuman and Jones polynomials (but not for the Homlfy polynomials) we get also a diierent proof for FPT via the corresponding result for signed Tutte polynomials.