islands outside the north and south caps. The remaining survey area in the genus measurement is 76.5% of the sky.

Genus Measurement
=================

We use the two-dimensional genus statistic introduced by Melott et al. (1989) and Gott et al. (1990) as a quantitative measure of topology of the CMB anisotropy field. For the two-dimensional CMB anisotropy temperature field, the genus is the number of hot spots minus the number of cold spots. Equivalently, the genus at a given threshold level $\nu$ is $$g(\nu) = {1 \over {2\pi}} \int_C \kappa ds,$$ where $\kappa$ is the signed curvature of the iso-temperature contours $C$, and the threshold level $\nu$ is the number of standard deviations from the mean. At a given threshold level, we measure the genus by integrating the curvature along iso-temperature contours. The curvature is positive (negative) if the interior of a contour has higher (lower) temperature than the specified threshold level. Compared to the CONTOUR2D algorithm (Melott et al. 1989) that was used in Colley & Gott (2003), this direct contour-integration method has an advantage that it can accurately calculate the genus when the survey region is enclosed by complicated boundaries (see Fig. 4$a$ below).

![Genus per steradian measured from 