In this paper, we focus on topological approaches to space and we argue that experiences with topology allow middle school students to develop a more robust understanding of orientation and dimension. We frame our argument in terms of the phenomenological literature on perception and corporeal space. We discuss findings from a quasi-experimental study engaging 9 grades 5-8 students (10-13 years old) in a 6-week series of school-based workshops focused on knot theory. We discuss video data that shows how students engage with the intrinsic disorientation of mathematical knots through the use of gesture and movement.