  A birack is an algebraic structure with axioms encoding the blackboard-framed
Reidemeister moves, incorporating quandles, racks, strong biquandles and
semiquandles as special cases. In this paper we extend the counting invariant
for finite racks to the case of finite biracks. We introduce a family of
biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles
and Silver-Williams switches, known as $(\tau,\sigma,\rho)$-biracks. We
consider enhancements of the counting invariant using writhe vectors, image
subbiracks, and birack polynomials.
