Using the Very Large Array, we have investigated a non-thermal radio filament (NTF) recently found very near the Galactic black hole and its radio counterpart, SgrA*. While this NTF -- the Sgr A West Filament (SgrAWF) -- shares many characteristics with the population of NTFs occupying the central few hundred parsecs of the Galaxy, the SgrAWF has the distinction of having an orientation and sky location that suggest an intimate physical connection to SgrA*. We present 3.3 and 5.5 cm images constructed using an innovative methodology that yields a very high dynamic range, providing an unprecedentedly clear picture of the SgrAWF. While the physical association of the SgrAWF with SgrA* is not unambiguous, the images decidedly evoke this interesting possibility. Assuming that the SgrAWF bears a physical relationship to SgrA*, we examine the potential implications. One is that SgrA* is a source of relativistic particles constrained to diffuse along ordered local field lines. The relativistic particles could also be fed into the local field by a collimated outflow from SgrA*, perhaps driven by the Poynting flux accompanying the black hole spin in the presence of a magnetic field threading the event horizon. Second, we consider the possibility that the SgrAWF is the manifestation of a low-mass-density cosmic string that has become anchored to the black hole. The simplest form of these hypotheses would predict that the filament be bi-directional, whereas the SgrAWF is only seen on one side of SgrA*, perhaps because of the dynamics of the local medium.
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