  We construct the skein category $\mathcal{S}$ of the annulus and show that it
is equivalent to the affine Temperley-Lieb category of Graham and Lehrer. It
leads to a skein theoretic description of the extended affine Temperley-Lieb
algebras. We construct an endofunctor of $\mathcal{S}$ that corresponds, on the
level of tangle diagrams, to the insertion of an arc connecting the inner and
outer boundary of the annulus. We use it to define and construct towers of
extended affine Temperley-Lieb algebra modules. It allows us to construct a
tower of modules acting on spaces of link patterns on the punctured disc which
play an important role in the study of loop models. In case of trivial Dehn
twist we show that the direct sum of the representation spaces of the link
pattern tower defines a graded algebra that may be regarded as a relative
version of the Roger-Yang skein algebra of arcs and links on the punctured
disc. We also describe the link pattern tower in terms of fused extended affine
Temperley-Lieb algebra modules.
