Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization

TL;DR

The paper defines the equivariant double-slice genus $\tilde{g}_{ds}(K,\tau)$ and the equivariant super-slice genus $\tilde{g}_{ss}(K,\tau)$ for strongly invertible knots, establishing lower bounds that separate these invariants from their non-equivariant and equivariant counterparts. It proves a key bound $\min\{g_{ds}(K_0), g_{ds}(K_1)\} \leq \tilde{g}_{ds}(K,\tau)$ and constructs families like $K_n$ that are double-slice and equivariantly slice yet have $\tilde{g}_{ds}(K_n,\tau) \ge n$, also producing symmetric unknotted 2-spheres that do not bound invariant symmetric 3-balls. The work connects these genera to stabilization, providing lower bounds for internal stabilization distance $d_1$ and introducing a symmetric stabilization distance $\tilde{d}_1^\tau$, with results extending to the equivariant setting. Additionally, the authors develop equivariant superslicing theory, and construct symmetrically knotted 2-spheres, illustrating new phenomena in equivariant knot and 2-knot theory that go beyond the classical (non-equivariant) framework. Overall, the paper provides a cohesive framework for obstructing symmetric isotopy and stabilization of surfaces via equivariant genera, with concrete constructions and implications for symmetric 2-knots and 2-spheres.

Abstract

In this paper we define the equivariant double-slice genus and equivariant super-slice genus of a strongly invertible knot. We prove lower bounds for both the equivariant double-slice genus and the equivariant super-slice genus. Using these bounds we find a family of knots which are double-slice and equivariantly slice, but have equivariant double-slice genus at least $n$. Using this result, we construct unknotted symmetric 2-spheres which do not bound symmetric 3-balls. Additionally, using double-slice and super-slice genera we find effective lower bounds for 1-handle stabilization distance and identify a possible method for using equivariant double-slice and super-slice genera to bound symmetric 1-handle stabilization distance for symmetric surfaces.

Figures (10)

• Figure 1: $(K_n,\tau)$
• Figure 2: $L_n$, i.e. $n$ copies of $8_{20}$ summed together.
• Figure 3: A strongly invertible knot ($9_{46},\tau,h$) and its antipode.
• Figure 4: The equivariant double of $8_{20}$.
• Figure 5: Example of vertical $H_i$ components (left) and horizontal $H_0$ components (right) with fixed point set in red.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 2.1: Orson and Powell
• Example 1
• Definition 2.2