To characterize the substructures induced in protoplanetary disks by the interaction between stars in multiple systems, we study the $1.25,$mm continuum and the $^{12}$CO$(J=2-1)$ spectral line emission of the triple systems HT Lup and AS 205, at scales of $approx 5,$au, as part of the Disk Substructures at High Angular Resolution Project (DSHARP). In the continuum emission, we find two symmetric spiral arms in the disk around AS 205 N, with pitch angle of $14^circ$, while the southern component AS 205 S, itself a spectroscopic binary, is surrounded by a compact inner disk and a bright ring at a radius of $34,$au. The $^{12}$CO line exhibits clear signatures of tidal interactions, with spiral arms, extended arc-like emission, and high velocity gas, possible evidence of a recent close encounter between the disks in the AS 205 system, as these features are predicted by hydrodynamic simulations of fly-by encounters. In the HT Lup system, we detect continuum emission from all three components. The primary disk, HT Lup A, also shows two-armed symmetric spiral structure with a pitch angle of $4^circ$, while HT Lup B and C, located at $25$ and $434,$au in projected separation from HT Lup A, are barely resolved with $sim5$ and $sim10,$au in diameter, respectively. The gas kinematics for the closest pair indicates a different sense of rotation for each disk, which could be explained by either a counter rotation of the two disks in different, close to parallel, planes, or by a projection effect of these disks with a close to $90^circ$ misalignment between them.
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