Weighted Interlace Polynomials

The interlace polynomials extend in a natural way to invariants of graphs with vertex-weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formula q(G) = q(G − a) + q(G − b) + ((x − 1) − 1)q(G − a − b) that lacks the last term; consequently the use of vertex-weights allows for interlace polynomial calculations represented by strictly binary computation trees rather than mixed binary-ternary computation trees. An intuitively pleasing property is that graphs which can be constructed through graph substitutions have vertex-weighted interlace polynomials which can be obtained through algebraic substitutions. Three other novel properties are weighted pendanttwin reductions, which involve removing certain kinds of vertices from a graph and adjusting the weights of the remaining vertices in such a way that the interlace polynomials are unchanged; these reductions allow for smaller computation trees as they eliminate some branches. If a graph can be completely analyzed using pendant-twin reductions then its interlace polynomial can be calculated in polynomial time, in a manner analogous to the calculation of the cumulative resistance of a collection of resistors wired in series and parallel. Vertex-weighted interlace polynomials also give a combinatorial description of the interlace polynomials of forests and trees, and they directly yield the Jones polynomials of classical links.