We studied the atomic structure of a multiple-twin triple junction in silicon, formed by the convergence of two (111) and one (221) symmetric-tilt grain boundaries. Molecular-dynamics simulations with the Stillinger-Weber potential and constant-traction border conditions were performed on several triple junction configurations, obtained by different combinations of the three grain boundaries. All the configurations have a positive excess line energy, a measurable volume contraction and display regions of opposite, tensile, and compressive, residual stress. We propose that triple junctions comprising twin boundaries could have the largest possible values of line tension and residual stress and discuss their role in the elastic stability of polycrystalline Si.