  Three-dimensional N=2 superconformal field theories are constructed by
compactifying M5-branes on three-manifolds. In the infrared the branes
recombine, and the physics is captured by a single M5-brane on a branched cover
of the original ultraviolet geometry. The branch locus is a tangle, a
one-dimensional knotted submanifold of the ultraviolet geometry. A choice of
branch sheet for this cover yields a Lagrangian for the theory, and varying the
branch sheet provides dual descriptions. Massless matter arises from vanishing
size M2-branes and appears as singularities of the tangle where branch lines
collide. Massive deformations of the field theory correspond to resolutions of
singularities resulting in distinct smooth manifolds connected by geometric
transitions. A generalization of Reidemeister moves for singular tangles
captures mirror symmetries of the underlying theory yielding a geometric
framework where dualities are manifest.
