  Motivated by circle graphs, and the enumeration of Euler circuits, we define
a one-variable ``interlace polynomial'' for any graph. The polynomial satisfies
a beautiful and unexpected reduction relation, quite different from the cut and
fuse reduction characterizing the Tutte polynomial.
  It emerges that the interlace graph polynomial may be viewed as a special
case of the Martin polynomial of an isotropic system, which underlies its
connections with the circuit partition polynomial and the Kauffman brackets of
a link diagram. The graph polynomial, in addition to being perhaps more broadly
accessible than the Martin polynomial for isotropic systems, also has a
two-variable generalization that is unknown for the Martin polynomial. We
consider extremal properties of the interlace polynomial, its values for
various special graphs, and evaluations which relate to basic graph properties
such as the component and independence numbers.
