We present high-resolution observations (0.2-1.5) of multiple dense gas tracers, HCN and HCO$^+$ ($J$ = 1-0, 3-2, and 4-3), HNC ($J$ = 1-0), and CS ($J$ = 7-6) lines, toward the nearby luminous infrared galaxy VV 114 with the Atacama Large Millimeter/submillimeter Array. All lines are robustly detected at the central gaseous filamentary structure including the eastern nucleus and the Overlap region, the collision interface of the progenitors. We found that there is no correlation between star formation efficiency and dense gas fraction, indicating that the amount of dense gas does not simply control star formation in VV 114. We predict the presence of more turbulent and diffuse molecular gas clouds around the Overlap region compared to those at the nuclear region assuming a turbulence-regulated star formation model. The intracloud turbulence at the Overlap region might be excited by galaxy-merger-induced shocks, which also explains the enhancement of gas-phase CH$_3$OH abundance previously found there. We also present spatially resolved spectral line energy distributions of HCN and HCO$^+$ for the first time, and derive excitation parameters by assuming optically-thin and local thermodynamic equilibrium (LTE) conditions. The LTE model revealed that warmer, HCO$^+$-poorer molecular gas medium is dominated around the eastern nucleus, harboring an AGN. The HCN abundance is remarkably flat ($sim$3.5 $times$ 10$^{-9}$) independently of the various environments within the filament of VV 114 (i.e., AGN, star formation, and shock).
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