  The three-dimensional topologies of the membrane of M-theory can be
constructed by performing Dehn surgery along knot lines. We investigate
membranes wrapped around a circle and the correponding subset of topologies
(Seifert manifolds). The knot lines are interpreted as magnetic flux tubes in
an XY model coupled to Maxwell theory. In this model the eleventh dimension of
M-theory gets ``eaten'' by the world-brane metric. There is argued to be a
second-order phase transition at a critical value of the string coupling
constant. The topology fluctuations that correspond to the knot lines are
irrelevant in one phase while they condense in the other phase.
