  Gel electrophoresis is a powerful experimental method to probe the topology
of DNA and other biopolymers. While there is a large body of experimental work
which allows us to accurately separate different topoisomers of a molecule, a
full theoretical understanding of these experiments has not yet been achieved.
Here we show that the mobility of DNA knots depends crucially and subtly on the
physical properties of the gel, and in particular on the presence of dangling
ends. The topological interactions between these and DNA molecules can be
described in terms of an "entanglement number", and yield a non-monotonic
mobility at moderate fields. Consequently, in two-dimensional electrophoresis,
gel bands display a characteristic arc pattern; this turns into a straight line
when the density of dangling ends vanishes. We also provide a novel framework
to accurately predict the shape of such arcs as a function of molecule length
and topological complexity, which may be used to inform future experiments.
