  A chord diagram consists of a circle, called the backbone, with line
segments, called chords, whose endpoints are attached to distinct points on the
circle. The genus of a chord diagram is the genus of the orientable surface
obtained by thickening the backbone to an annulus and attaching bands to the
inner boundary circle at the ends of each chord. Variations of this
construction are considered here, where bands are possibly attached to the
outer boundary circle of the annulus. The genus range of a chord diagram is the
genus values over all such variations of surfaces thus obtained from a given
chord diagram. Genus ranges of chord diagrams for a fixed number of chords are
studied. Integer intervals that can, and cannot, be realized as genus ranges
are investigated. Computer calculations are presented, and play a key role in
discovering and proving the properties of genus ranges.
