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Rotationally-resolved Fourier-transform spectra of laser-induced fluorescence(LIF) $A^1\Sigma^+_u\sim b^3\Pi_u \rightarrow X^1\Sigma^+_g$ of K$_2$ moleculeswere recorded and analyzed yielding 4053 term values of spin-orbit (SO) coupled$A \sim b$ complex of $^{39}$K$_2$ isotopologue with ca 0.01 cm$^{-1}$accuracy. Their compilation with 1739 term values from previously publishedsources allowed to cover the energy range [9955, 17436] cm$^{-1}$ from thebottom of the lower-lying $b^3\Pi_u$ state up to the vicinity of the atomicasymptote $4s\;^2{\rm S}_{1/2}$ + $4p\;^2{\rm P}_{1/2}$, with rotationalquantum number $J\in[0,149]$. The experimental data were processed by a direct6$\times$6 coupled-channel (CC) deperturbation treatment, which accountedexplicitly for both SO and electronic-rotational interactions between sixstates: $A^1\Sigma^+_u$($0$), $b^3\Pi_u$(0,1,2), $c^3\Sigma_u$(1), and$B^1\Pi_u$(1). The initial parameters of the global deperturbation model havebeen estimated in the framework of {\it ab initio} electronic structurecalculations applying multi-referenced configuration-interaction andcoupled-clusters methods. The interatomic potentials analytically defined for$A$ and $b$ states, as well as SO-splitting of the triplet $b$ state and $A\simb$ SO-coupling functions have been refined to fit 5792 term values of the$^{39}$K$_2$ isotopologue, whereas the rest parameters were fixed on their {\itab initio} values. The resulting mass-invariant parameters of the CC modelreproduced the overall rovibronic term energies of the $A \sim b$ complex of$^{39}$K$_2$ with the accuracy, which is well within the experimental errors.This deperturbation analysis provided the refined dissociation energy$T_{dis}$=17474.569(5) cm$^{-1}$ and the long-range coefficient $C_3^{\Sigma}$= 5.501(4) $\times10^5$cm$^{-1}$ A$^3$ relevant to the non-relativistic atomiclimit $4s$+$4p$.

Fourier-transform spectroscopy and global deperturbation treatment of  the $A^1\Sigma^+_u$ and $b^3\Pi_u$ states of K$_2$ in the entire bound energy  range