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Measurements of Absolute Branching Fractions of Fourteen Exclusive Hadronic $D$ Decays to $η$

Using $2.93\,\rm fb^{-1}$ of $e^+e^-$ collision data taken at a center-of-mass energy of 3.773\,GeV with the BESIII detector, we report the first measurements of the absolute branching fractions of fourteen hadronic $D^{0(+)}$ decays to exclusive final states with an $η$, e.g., $D^0\to K^-π^+η$, $K^0_Sπ^0η$, $K^+K^-η$, $K^0_SK^0_Sη$, $K^-π^+π^0η$, $K^0_Sπ^+π^-η$, $K^0_Sπ^0π^0η$, and $π^+π^-π^0η$; $D^+\to K^0_Sπ^+η$, $K^0_SK^+η$, $K^-π^+π^+η$, $K^0_Sπ^+π^0η$, $π^+π^+π^-η$, and $π^+π^0π^0η$. Among these decays, the $D^0\to K^-π^+η$ and $D^+\to K^0_Sπ^+η$ decays have the largest branching fractions, which are $\mathcal{B} (D^0\to K^-π^+η)=(1.853\pm0.025_{\rm stat}\pm0.031_{\rm syst})\%$ and $\mathcal{B}(D^+\to K^0_Sπ^+η)=(1.309\pm0.037_{\rm stat}\pm0.031_{\rm syst})\%$, respectively. We also determine the $CP$ asymmetries for the six decays with highest event yields. No evidence of $CP$ violation is found.