The smooth (or topological) 4-genus of a knot is the minimum genus of a smooth (topological, locally-flat) surface embedded in the 4-ball with boundary the link.
For knots, some bounds are determined by the p-signatures and, to avoid being of 4-genus 0 (slice), the Alexander polynomial. In addition, there are bounds determined by gauge theoretic invariants, which apply only in the smooth category. The knots for which these smooth techniques are required are marked in the table with reference links.
For multi-component links, the relevant lower bounds are determined by the Arf-Robertello invariant, the signature, and in the smooth setting the Slice-Bennequin bound.
All the data about the four-genus presented in LinkInfo was computed by Stepan Orevkov. He computed the upper bounds by performing crossing changes in the minimal diagrams for the links.
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