Seifert surfaces in the 4-ball

Abstract

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in S 3 that do not become isotopic when their interiors are pushed into B 4 . We give examples where the surfaces are not topologically isotopic in B 4 , as well as examples that are topologically but not smoothly isotopic. These latter surfaces are distinguished by their associated cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.

Invariants of 4–manifolds from Khovanov–Rozansky link homology

“…Here we make this idea precise and explain how a smoothly embedded oriented surface 𝑆 determines an element of tridegree depend on the Euler characteristic 𝜒 (𝑆) and the self-intersection of 𝑆 computed with respect to the pushoff determined by the framing of 𝐿 along the boundary. When 𝑊 = 𝐵 4 and 𝑁 = 2, the surface invariant is given by the relative Khovanov-Jacobsson classes [SS22] or, in other words, by cobordism maps in Khovanov homology, which are known to detect exotica [HS21;Hay+23].…”