A collinear beams apparatus has been used to determine photodetachment cross sections for K − in the photon energy range 4.250-4.360 eV. State-selective detection, utilizing a resonance ionization scheme, was applied to measure partial cross sections for those channels which leave the residual K atoms in the excited 7 2 S, 5 2 F , and 5 2 G states. The energy region studied encompassed the openings of the aforementioned channels, as well as the channel that leaves the K atom in the 7 2 P state. Two previously unobserved resonances were seen in all three partial cross sections between the K(5 2 G) and K(7 2 P ) thresholds. It is shown that a more reliable determination of resonance parameters can be made if the same resonances are observed in several channels. In the region below the K(5 2 F ) threshold, three previously observed resonances were investigated [Kiyan et al., Phys. Rev. Lett. 84, 5979 (2000)]. A greatly increased modulation of the signal was obtained by detecting in the K(7 2 S) channel instead of the K(5 2 S) channel used in the previous study. Furthermore, the shapes of the cross sections in the threshold regions are discussed. A detailed description of the apparatus and the experimental procedure employed is presented in the paper.