Studies of $e^+e^- to D_s^+ overline{D}{}^{(*)0}K^-$ and the $P$-wave charmed-strange mesons are performed based on an $e^+e^-$ collision data sample corresponding to an integrated luminosity of 567 pb$^{-1}$ collected with the BESIII detector at $sqrt{s}= 4.600$ GeV. The processes of $e^+e^-to D_s^+ overline{D}{*}^{0} K^-$ and $D_s^+ overline{D}{}^{0} K^-$ are observed for the first time and are found to be dominated by the modes $D_s^+ D_{s1}(2536)^-$ and $D_s^+ D^*_{s2}(2573)^-$, respectively. The Born cross sections are measured to be $sigma^{B}(e^+e^-to D_s^+ overline{D}{*}^{0} K^-) = (10.1pm2.3pm0.8) pb$ and $sigma^{B}(e^+e^-to D_s^+ overline{D}{}^{0} K^-) = (19.4pm2.3pm1.6) pb$, and the products of Born cross section and the decay branching fraction are measured to be $sigma^{B}(e^+e^-to D_s^+D_{s1}(2536)^- + c.c.)cdotmathcal{B}(D_{s1}(2536)^- to overline{D}{*}^{0} K^-) = (7.5 pm 1.8 pm 0.7) pb$ and $sigma^{B}(e^+e^-to D_s^+ D^*_{s2}(2573)^- + c.c.)cdotmathcal{B}(D^*_{s2}(2573)^- to overline{D}{}^{0} K^-) = (19.7 pm 2.9 pm 2.0) pb$. For the $D_{s1}(2536)^-$ and $D^*_{s2}(2573)^-$ mesons, the masses and widths are measured to be $M(D_{s1}(2536)^-) = (2537.7 pm 0.5 pm 3.1)~MeV/c^2,$ $ Gamma(D_{s1}(2536)^-)) = (1.7pm 1.2 pm 0.6)~rm MeV,$ and $M(D^*_{s2}(2573)^-) = (2570.7pm 2.0 pm 1.7)~MeV/c^2,$ $Gamma(D^*_{s2}(2573)^-) = (17.2 pm 3.6 pm 1.1)~rm MeV.$ The spin-parity of the $D^*_{s2}(2573)^-$ meson is determined to be $J^P=2^{+}$. In addition, the process $e^+e^-to D_s^+ overline{D}{}^{(*)0} K^-$ are searched for using the data samples taken at four (two) center-of-mass energies between 4.416 (4.527) and 4.575 GeV, and upper limits at the $90%$ confidence level on the cross sections are determined.
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