Knotting and linking in 4 and 5 dimensions from barbell diffeomorphisms

Abstract

In this paper, we construct infinitely many non-isotopic 3-knots in the 5-sphere, each of which has four critical points with respect to the standard height function of the 5-sphere. This contrasts with a theorem of Scharlemann which says that any 2-knot in the 4-sphere with four critical points is unknotted, and also provides infinitely many knotted solid tori in the 4-sphere and 5-ball, which resolves the last remaining case of the conjecture by Budney and Gabai on the existence of knotted handlebodies. We also construct various knotted and linked handlebodies, discs, and spheres in the 4-sphere, 5-ball, and 5-sphere, extending recent works of Hughes, Miller, and the first author, and a recent work of the authors. All of our examples are explicit and are constructed using barbell diffeomorphisms.

Figures (15)

• Figure 2.1: (a): The arcs $A_{1}$ and $A_{2}$, the 2-spheres $S_{1}$ and $S_{2}$, and the 2-disks $D_{1}$ and $D_{2}$ in $B^{3}\times0 \subset B^4$. Note that $D_{1}$ appears fully in this 3-dimensional slice, while $D_{2}$ intersects it in an arc; similarly, $S_{2}$ appears fully in this slice, while $S_{1}$ intersects it in an equatorial $S^{1}$. (b): The model barbell $\mathcal{B}$.
• Figure 3.1: Circles $C_{L}$ and $C_{R}$, disks $D_{L}$ and $D_{R}$, a simple barbell $\beta$ with cuffs $S_{L}$ and $S_{R}$, and a splitting 3-sphere $\Sigma$ in $S^{4}$; their intersections with the $S^{3}\times0$ time slice are drawn. Note that only the equatorial $S^{1}$’ s of the cuffs of the barbell, and the equatorial $S^{2}$ of the splitting sphere $\Sigma$ are drawn.
• Figure 3.2: The standard genus $g$ surface $K_g \subset S^3 \times 0 \subset S^4$, the horizontal handlebody $H_h \subset S^3 \times 0 \subset S^4$ that $K_g$ bounds, $2$-spheres $S_{h,i},S_{v,i} \subset S^4 \setminus \mathring {N}(K_g)$ for $i=1,\cdots ,g$ whose homology classes generate $H_{2}(S^{4}\setminus K_{g}) \cong\mathbb{Z}^{2g}$, and a compressing disk $D_h$ for the handlebody $H_h$ that is geometrically dual to $S_{h,1}$.
• Figure 3.3: The $2$-component unlink of two tori $K_{1,1} = K_{L}\sqcup K_{R}\subset S^{3}\times0$, a simple barbell $\beta$ with cuffs $S_L$ and $S_R$, a compressing disk $D_{R}$ for the horizontal handlebody that $K_{R}$ bounds, and a splitting $3$-sphere $\Sigma$.
• Figure 3.4: The standard genus $2$ surface $K_2 \subset S^3 \times 0 \subset S^4$, a barbell $\beta$ that gives rise to knotted handlebodies $\boldsymbol{\beta}^k H_h$, and a compressing disk $D_h$ for the handlebody $H_h$

Theorems & Definitions (79)

• Theorem 1.1: "Morse-simple" 3-knots; Theorem \ref{['thm:morsesimple-s3']}
• Definition 1.2: Compressing-curve equivalent handlebodies
• Corollary 1.3: Knotted handlebodies; Theorem \ref{['thm:genus1-handlebody']} and Corollary \ref{['cor:bgconj']}
• Remark 1.4: Comparison with hughes2024knotted
• Theorem 1.5: Brunnian 3-links; Theorem \ref{['thm:linked-6crit']}
• Corollary 1.6: Brunnian handlebody links; Corollary \ref{['cor:solid-torus-ball-brunnian']}
• Remark 1.7: Brunnian $2$-disk links
• Definition 1.8: Splitting spheres
• Theorem 1.9: Knotted splitting spheres
• Remark 1.10: $4$-dimensional methods for knotted handlebodies