Legendrian Satellites and Decomposable Cobordisms

Abstract: We investigate the interactions between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots. Given Lagrangian cobordisms between two Legendrian knots and between two Legendrian tangles, we construct a Lagrangian cobordism between Legendrian satellites of the knots by the closures of the tangles, with extra twists on both the top and the bottom satellite to compensate for the genus of the cobordism. If the original cobordisms were decomposable, then a decomposable cobordism between satellites exists as well, again with extra twists.