Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer we find a pair of 2-knots in the 4-sphere whose stabilization distance equals .
Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer we exhibit a knot in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals~. We show this using homology of cyclic covers.
Finally, we use metabelian twisted homology to show that for each~ there exists a knot and pair of slice discs with generalized stabilization distance at least , with the additional property that abelian invariants associated to cyclic covering spaces coincide. This detects different choices of slicing discs corresponding to a fixed metabolising link on a Seifert surface.
Cite this article
Allison N. Miller, Mark Powell, Stabilization distance between surfaces. Enseign. Math. 65 (2019), no. 3/4, pp. 397–440DOI 10.4171/LEM/65-3/4-4