A statistical analysis of texture on the COBE-DMR first-year sky maps based on the genus and spot number is presented. A generalized |$\chi^2$| statistic is defined in terms of 'observable' quantities: the genus and spot density that would be measured by different cosmic observers. This strategy, together with the use of Monte Carlo simulations of the temperature fluctuations, including all the relevant experimental parameters, represents the main difference from previous analyses. Based on the genus analysis we find a strong anticorrelation between the quadrupole amplitude |$Q_\text{rms-PS}$| and the spectral index n of the density fluctuation power spectrum at recombination of the form |$Q_\text {rms-PS}=22.2\pm1.7-(4.7\pm1.3)\times n\mu\text K$| for fixed n, consistent with previous works. The result obtained based on the spot density is consistent with this |$Q_\text{rms-PS}(n)$| relation. In addition to the previous results, we have determined, using Monte Carlo simulations, the minimum uncertainty due to cosmic variance for the determination of the spectral index with the genus analysis. This uncertainty is |$\delta n\approx 0.2$|⁠.