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Although Venus appears to present a predominantly ionospheric obstacle to the solar wind, the magnetic connectivity between the solar wind and the Venus ionosphere, or magnetic topology, is important for characterizing the Venus space environment. In particular, magnetic connectivity is relevant to the magnetization state of the ionosphere, particle precipitation into the atmosphere causing ionization and auroral emissions, and planetary ion escape at Venus. The spatial distributions of different magnetic topologies were statistically analyzed, with some unexpected results. Here, we build on those results by investigating how the external factors of solar cycle phase and upstream conditions affect the occurrence rates of the three magnetic topologies and consider their implications regarding the state of Venus's induced magnetosphere. We find that both the solar cycle phase and upstream dynamic pressure variations control its expansion or contraction. Under solar minimum conditions, the interplanetary magnetic field (IMF) more deeply penetrates into the collisional atmosphere, increasing the occurrence rates of open and closed topologies at low altitudes and in Venus's wake. We also find hemispheric differences in the occurrences of dayside-connected and nightside-connected open fields, likely related to mass loading of the near-Venus plasma environment by planetary pickup ions.