On BV Homeomorphisms

We obtain the rectifiability of the graph of a bounded variation homeomorphism f in the plane and relations between gradients of f and its inverse. Further, we show an example of a bounded variation homeomorphism f in the plane which satisfies the (N) and $(N^{-1})$ properties and strict positivity of Jacobian of both itself and its inverse, but neither f nor $f^{-1}$ is Sobolev.