Smooth and Topological Nonorientable Four-Genera of Low-Crossing Knots

Abstract

We compare linking-form bounds with the smooth and locally flat nonorientable 4-genera of low-crossing knots. For every prime knot with at most 13 crossings whose Witt-level linking-form bound is at least 2, this bound equals both genera. We also exhibit 41 prime knots with 12 or 13 crossings that bound locally flat Möbius bands but no smooth ones, and whose smooth nonorientable 4-genus is 2. Our methods combine linking forms, Heegaard Floer correction terms and definite-lattice obstructions with explicit band constructions. They determine 3,300 of the 3,525 values left unknown in KnotInfo on 2 November 2026; the remaining 225 knots have range $[1,2]$. We obtain new lower bounds of 2 for 2,102 knots with 14 crossings and correct previously reported values for three knots with 11 crossings.