Higher-order analogues of genus and slice genus of classical knots
Horn, Peter Douglas
Cochran, Tim D.
Doctor of Philosophy
We define invariants analogous to the genus and slice genus of knots in S3. For algebraically slice, genus one knots, we define the differential genus, denoted dg, and we prove it is independent of the Alexander polynomial and knot Floer homology. For knots which bound Gropes of height n + 2 in D 4, we define the nth-order genus, denoted gn. Each of the n th-order genera is a generalization of the slice genus. For each n ≥ 1, we construct knots with identical lower-order genera and distinct nth-order genera, thus proving that these invariants are independent of one another. Finally, we employ the higher-order genera to give a refinement of the Grope filtration of the knot concordance group.