  A knot diagram has an associated looped interlacement graph, obtained from
the intersection graph of the Gauss diagram by attaching loops to the vertices
that correspond to negative crossings. This construction suggests an extension
of the Kauffman bracket to an invariant of looped graphs, and an extension of
Reidemeister equivalence to an equivalence relation on looped graphs. The graph
bracket polynomial can be defined recursively using the same pivot and local
complementation operations used to define the interlace polynomial, and it
gives rise to a graph Jones polynomial that is invariant under the graph
Reidemeister moves.
