  A determinant in algebraic $K$-theory is associated to any two almost
commuting Fredholm operators. On the other hand, one can calculate a
homologically defined invariant known as joint torsion. We answer in the
affirmative a conjecture of Richard Carey and Joel Pincus, namely that these
two invariants agree. In particular, this implies that joint torsion is norm
continuous, depends only on the images of the operators modulo trace class, and
satisfies the expected Steinberg relations. Moreover, we show that the
determinant invariant of two commuting operators can be computed simply as a
determinant on a finite dimensional vector space.
