A Dunfield--Gong 4-Sphere is Standard

Abstract

In this paper, we standardize a homotopy $4$-sphere constructed by Dunfield and Gong. As a corollary, we show that the $18$-crossing knot $18_{\text{nh}00000601}$, which is not known to be ribbon, is slice in the standard $4$-ball. Thus, $18_{\text{nh}00000601}$ serves as a potential counterexample to the Slice-Ribbon Conjecture. In addition, we show that the same knot bounds a fibered handle-ribbon disk in $B^{4}$.

Figures (9)

• Figure 1: Two knots, $K_{G}$ and $K_{B}$, constructed by Dunfield and Gong. The knot $K_{G}$ is denoted $18_{\text{nh}00000601}$ by Dunfield and Gong.
• Figure 2: (Left) $K_{B}$ pictured as a banded unlink with band specified in pink. (Right) We zoom in on our band used to express $K_{B}$ as a banded unlink.
• Figure 3: Dunfield and Gong's RBG link $R\cup B\cup G$ as pictured in Figure $19$ of DG with respective framings $r = 1, b = 0$, and $g = 0$.
• Figure 4: The link which depicts $K_{B}$ sitting inside the RBG-link after handlesliding $B$ over $R$.
• Figure 5: (Left) A Kirby diagram of $X_{DG}$ obtained from the link $K_{B}\cup R$ by dotting $K_{B}$, adding a $1$-framed $2$-handle to $R$, then capping off with a $4$-ball. (Right) A Kirby diagram for $X_{DG}$ obtained by performing a ribbon move on the dotted component in the figure on the left.

Theorems & Definitions (31)

• Theorem 1.1
• Corollary 1.1.1
• Corollary 1.1.2
• Theorem 1.2
• Definition 2.1
• Remark 2.1
• Theorem : Theorem \ref{['thm:mainthm']}
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm:mainthm']}