  The celebrated Aharonov-Bohm effect is perhaps the first example in which the
the interplay between classical topology and quantum theory was explored. This
connection has continued to shed light on diverse areas of physics like quantum
statistics, anomalies, condensed matter physics, and gauge theories. Several
attempts were made to generalize the Aharonov-Bohm effect by modifying the
simple solenoidal current distribution used by them to the case of multiple
solenoids, and a toroidal solenoid, for example. A particularly ambitious task
is to confine the magnetic flux to the interior of a knotted solenoid. While it
is to be expected that a non-trivial phase factor will be picked up by the wave
function of a charged particle travelling in the complement of the knot in
three-dimensional space, the lack of symmetry defied attempts to explicitly
solve the associated scattering problem. In this paper we report on a way to
make progress towards this problem based on multipole expansions. The vector
potential produced by a knot is obtained by making a multipole expansion, which
is then used to calculate the S-matrix for the scattering of the charged
particle, in the Born approximation. It is found that the S-matrix carries an
imprint of the knottedness at the octopole order. For the case of a torus knot,
a curious factorization property is seen to hold.
