Indecomposable Klein bottles with order-4 meridians

TL;DR

This work constructs an infinite family of indecomposable Klein bottles $\mathcal{K}(l,m,n)$ in $S^4$ with meridians of order $4$, built from a 3-stranded pretzel knot $K=P(l,m,n)$ via ribbon annuli and a flip-spin mapping-torus. It provides explicit group-theoretic descriptions, including the presentation $\pi(\mathcal{K}(l,m,n))=\langle a,b,c\mid a^4=b^4=c^4=(bc)^m=(ca)^n=1,(ab)^l=a^2=b^2=c^2\rangle$ and the branched-double-cover group $\pi_1(\Sigma_2(\mathcal{K}))=\langle u,v\mid u^l=v^m=(uv)^n=1\rangle$, enabling obstructions to decomposability via stable irreducibility arguments. For $m=n$ the authors adapt Lidman–Piccirillo's stable-irreducibility obstruction to show indecomposability, relying on signature-zero of $\Sigma_2(\mathcal{K})$, non-splitting of the von Dyck-type group, and amphicheirality; they further show that many $\mathcal{K}(l,m,n)$ are pairwise non-isotopic when the triples differ in absolute value. The work situates these examples among known indecomposable Klein bottles and raises open questions about ribbon-ness, potential disoriented-tube constructions, and whether sign-data affects isotopy.

Abstract

We exhibit an infinite family of indecomposable Klein bottles in the 4-sphere with order-4 meridians.

Figures (3)

• Figure 1: The Klein bottle $\mathcal{K}$
• Figure 2: Constructing the Klein bottles
• Figure 3: Three isotopic presentations of the Klein bottle $\mathcal{K}$.

Theorems & Definitions (20)

• Theorem 1
• Proposition 2
• Definition 3
• Remark 4
• Proposition 5
• proof
• Proposition 6
• proof
• Corollary 7
• proof