We give a flexible construction for knots in the 3-sphere that bound surfaces of unexpectedly low genus in punctured open books on 3-manifolds. We use this construction to give the first examples of knots whose genus differs in different $\mathbb{ Z}/2\mathbb{ Z}$ homology balls. We also establish that every knot bounds a M{ \"o}bius band in a rational homology ball, and that there are knots whose genus in $T^4$ and $B^4$ differ arbitrarily.
Categories: math.GT