Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space

Abstract: We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimal genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology 3-sphere and surfaces in certain simply connected signature zero 4-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched covering of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched covering is determined up to isometry by the action of the deck transformation group on said first homology. As an application of the new upper and lower bounds, we complete the calculation of the Z-slice genus for all prime knots with crossing number up to 12.