JournalslemVol. 65, No. 3/4pp. 397–440

Stabilization distance between surfaces

  • Allison N. Miller

    Rice University, Houston, USA
  • Mark Powell

    Durham University, UK
Stabilization distance between surfaces cover

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Abstract

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 mm we find a pair of 2-knots in the 4-sphere whose stabilization distance equals mm.

Next, using a generalized stabilization distance that counts connected sum with arbitrary 2-knots as distance zero, for every nonnegative integer mm we exhibit a knot JmJ_m in the 3-sphere with two slice discs in the 4-ball whose generalized stabilization distance equals~mm. We show this using homology of cyclic covers.

Finally, we use metabelian twisted homology to show that for each~mm there exists a knot and pair of slice discs with generalized stabilization distance at least mm, 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 (2020), no. 3, pp. 397–440

DOI 10.4171/LEM/65-3/4-4