We report on high resolution Fourier-transform spectroscopy of fluorescence to the a^3Sigma_u^+ state excited by two-photon or two-step excitation from the X^1Sigma_g^+ state to the 2^3Pi_g state in the molecule K_2. These spectroscopic data are combined with recent results of Feshbach resonances and two-color photoassociation spectra for deriving the potential curves of X^1Sigma_g^+ and a^3Sigma_u^+ up to the asymptote. The precise relative position of the triplet levels with respect of the singlet levels was achieved by including the excitation energies from the X^1Sigma_g^+ state to the 2^3Pi_g state and down to the a^3Sigma_u^+ state in the simultaneous fit of both potentials. The derived precise potential curves allow for reliable modeling of cold collisions of pairs of potassium atoms in their ^2S ground state.
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