Smooth Four-Genus

The smooth 4-genus of a knot is the minimum genus of a smooth surface embedded in the 4-ball with boundary the knot. Bounds arise from the p-signatures and, to obstruct being of 4-genus 0 (slice) the Alexander polynomial. In addition, there are bounds arising from gauge theoretic invariants, and these apply only in the smooth category. Those knots for which these smooth techniques are required are marked in the table with reference links.

Further information on particular knots.