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Higher-order Genera of Knots

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higherorder genera in terms of certain von Neumann ρ-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.