  Let $D$ be an oriented classical or virtual link diagram with directed
universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in
each connected component of $U$. There is then an associated looped
interlacement graph $L(D,C)$ whose construction involves very little geometric
information about the way $D$ is drawn in the plane; consequently $L(D,C)$ is
different from other combinatorial structures associated with classical link
diagrams, like the checkerboard graph, which can be difficult to extend to
arbitrary virtual links. $L(D,C)$ is determined by three things: the structure
of $\vec{U}$ as a 2-in, 2-out digraph, the distinction between crossings that
make a positive contribution to the writhe and those that make a negative
contribution, and the relationship between $C$ and the directed circuits in
$\vec{U}$ arising from the link components; this relationship is indicated by
marking the vertices where $C$ does not follow the incident link component(s).
We introduce a bracket polynomial for arbitrary marked graphs, defined using
either a formula involving matrix nullities or a recursion involving the local
complement and pivot operations; the marked-graph bracket of $L(D,C)$ is the
same as the Kauffman bracket of $D$. This provides a unified combinatorial
description of the Jones polynomial that applies seamlessly to both classical
and non-classical virtual links.
