On the detection of knotted spheres by their traces in high dimensions

TL;DR

The paper proves that in dimensions $n\ge4$ there exist non-isotopic knotted $(n-2)$-spheres in $S^n$ whose traces are orientation-preservingly diffeomorphic, by generalising the RBG construction to all dimensions. It further shows that the unknot is detected by its surgery (and hence by its trace) in these high dimensions, while also presenting infinite families of distinct knots sharing traces via Gluck twists and Plotnick-type constructions. A key innovation is distinguishing knots with the same trace using counts of representations of their knot groups into a finite group (notably $A_5$), and, where necessary, employing Fox calculus to compare Alexander polynomials. Collectively, these results delineate the limitations of trace-surgery data for detecting knot isotopy in high dimensions and illuminate the interplay between traces, Gluck twists, and higher-dimensional knot theory.

Abstract

For every $n \geq 4$, we demonstrate the existence of non-isotopic smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces by generalising the RBG link construction to all dimensions. Conversely, we prove that for every $n \geq 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence by its trace.

Figures (5)

• Figure 1: Construction of the knot $B$ in $S^2 \times S^{n-2}$ as connected sum of a local knot $K$ and $\{w\} \times S^{n-2}$. The belt sphere $\mathcal{B}_R = S^2 \times \{z\}$ is in red.
• Figure 2: Choice of arcs in $S^2 \times \{z\}$ that give rise to "straight" arcs $\beta_1, \dots, \beta_{2\ell}$ in $S^2 \times S^{n-2}$. The knot $\widetilde{G}$ in $S^2 \times S^{n-2}$ is obtained by tubing $\bigcup \{x_i\} \times S^{n-2}$ along the arcs $\beta_i$.
• Figure 3: Band sum $\beta_i\#_{\delta_i} \gamma_i$ of the arc $\beta_i$ and $\gamma_i \colon S^1 \hookrightarrow S^n \setminus K$ along an arc $\delta_i$. On the right, the region giving an isotopy between $\beta_i\#_{\delta_i} \gamma_i$ and $\beta_i$ in $S^2 \times S^{n-2}$ is shaded.
• Figure 4: A schematic for an isotopy between $\widetilde{G}$ and $\{x_{2\ell-1}\}\times S^{n-2}$.
• Figure 5: On the left: Discs $D^g_m$ bounded by $g_m$ (in green) and $D^b_m$ bounded by $b_m$ (in blue) in $S^2\times D^2$ for $n=4$. On the right: Kirby diagram for $S^2\times S^2\setminus (B_m\cup G_m)$.

Theorems & Definitions (45)

• Theorem 1
• Remark 1.1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4