  Knitted and woven textile structures are examples of doubly periodic
structures in a thickened plane made out of intertwining strands of yarn.
Factoring out the group of translation symmetries of such a structure gives
rise to a link diagram in a thickened torus. Such a diagram on a standard torus
is converted into a classical link by including two auxiliary components which
form the cores of the complementary solid tori. The resulting link, called a
kernel for the structure, is determined by a choice of generators u and v for
the group of symmetries.
  A normalised form of the multi-variable Alexander polynomial of a kernel is
used to provide polynomial invariants of the original structure which are
essentially independent of the choice of generators. It gives immediate
information about the existence of closed curves and other topological features
in the original textile structure. Because of its natural algebraic properties
under coverings we can recover the polynomial for kernels based on a proper
subgroup from the polynomial derived from the full symmetry group of the
structure. This enables two structures to be compared at similar scales, even
when one has a much smaller minimal repeating cell than the other.
  Examples of simple traditional structures are given, and their Alexander data
polynomials are presented to illustrate the techniques and results.
