  The algebraic genus of a knot is an invariant that arises when one considers
upper bounds for the topological slice genus coming from Freedman's theorem
that Alexander polynomial one knots are topologically slice. This paper
develops null-homologous twisting operations as a tool for studying the
algebraic genus and, consequently, for bounding the topological slice genus
above. As applications we give new upper bounds on the algebraic genera of
torus knots and satellite knots.
