  The involutory birack counting invariant is an integer-valued invariant of
unoriented tangles defined by counting homomorphisms from the fundamental
involutory birack of the tangle to a finite involutory birack over a set of
framings modulo the birack rank of the labeling birack. In this first of an
anticipated series of several papers, we enhance the involutory birack counting
invariant with quantum weights, which may be understood as tangle functors of
involutory birack-labeled unoriented tangles.
