Unknotting numbers of 2‐spheres in the 4‐sphere

Abstract

We compare two naturally arising notions of ‘unknotting number’ for 2‐spheres in the 4‐sphere: namely, the minimal number of 1‐handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2‐sphere. We refer to these invariants as the stabilization number and the Casson–Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson–Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non‐additivity, and applications to classical unknotting number of 1‐knots.

Invariants of $2$-knots

“…Further invariants can be extracted from the knot group. For example, the meridional rank [64] is the minimal number of meridians that generate π 1 (X K ), whereas the Ma-Qiu index [94] is the minimal size of a normal generating set of the commutator subgroup of π 1 (X K ), the weak unknotting number (referred to as the algebraic stabilisation number in [63]) is the minimal number of relations of the form x = y, where x and y are meridians of K, which abelianise π 1 (X K ) [69]. These latter two invariants can be used to give lower bounds on the unknotting number of a 2-knot K [60].…”

“…These latter two invariants can be used to give lower bounds on the unknotting number of a 2-knot K [60]. Here the unknotting number of K (referred to as the stabilisation number in [63]) is the minimal number of stabilisations of K ⊂ S 4 required to obtain an unknotted surface (every 2-knot K ⊂ S 4 has finite unknotting number [59]; see [10,6] for related results and [24, Appendix A] for the details in the topological category). The Alexander module H 1 ([π 1 (X K ), π 1 (X K )]) = π 1 (X K ) (1) /π 1 (X K ) (2) (and the resulting Alexander invariants) is discussed in Section 3 but we already record that it can used to produce lower bounds on the stabilisation number [98,97,63].…”

“…The k-th Alexander polynomial ∆ (k) K of a 2-knot K refers to the greatest common divisor of its k-th Alexander ideal; we refer to [90] for more properties of these polynomials. Further Alexander invariants include the Nakanishi index of a 2-knot (the minimal number of generators of H 1 (X ∞ K ) [101,63]) and the determinant of a 2-knot (the evaluation of the Alexander ideal of K at t = −1 [62]). All of these Alexander invariants can be generalised to surface knots of higher genus.…”

Knotted handlebodies in the 4-sphere and 5-ball

“…Joseph-Klug-Ruppik-Schwartz [5] gave examples of when the tubings prescribed by Lemma 3.2 are particularly simple.…”