This paper contains the results of efforts to determine values of the smooth
and the topological slice genus of 11- and 12-crossing knots. Upper bounds for
these genera were produced by using a computer to search for genus one
concordances between knots. For the topological slice genus further upper
bounds were produced using the algebraic genus. Lower bounds were obtained
using a new obstruction from the Seifert form and by use of Donaldson's
diagonalization theorem. These results complete the computation of the
topological slice genera for all knots with at most 11 crossings and leaves the
smooth genera unknown for only two 11-crossing knots. For 12 crossings there
remain merely 25 knots whose smooth or topological slice genus is unknown.Comment: 9 pages + 11 pages of appendices. This is a substantial expansion of
the original article. This version features a second author and new
techniques for calculating the topological slice genu