Non-complex cobordisms between quasipositive knots

Abstract

We show that for every genus $g\ge 0$, there exist quasipositive knots $K_0^g$ and $K_1^g$ such that there is a cobordism of genus $g=|g_4(K_1^g)-g_4(K_0^g)|$ between $K_0^g$ and $K_1^g$, but there is no ribbon cobordism of genus $g$ in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016.

13 pages, 5 figures, 1 table. Comments welcome! v3: More details and improved exposition following the referee's comments. Section 5 added, along with 5 figures. Corresponds to version accepted for publication in J. Math. Pures Appl