On the connected sums of the $(2,1)$-cable of the figure eight knot

TL;DR

This paper proves that the 3-fold and 6-fold connected sums of the $(2,1)$-cable of the figure-eight knot cannot bound smooth null-homologous disks in punctured connected sums of $S^2\times S^2$, by employing a real version of the $10/8$ inequality within a $\mathbb{Z}_2$-equivariant framework. Central to the argument is a smooth concordance from the figure-eight to a slice knot in a twice-punctured $2\mathbb{C}P^2$ representing $(1,3)$ in $H_2(X,\partial X;\mathbb{Z})$, which enables the application of the real $10/8$-inequality to branched-cover cobordisms. The results yield lower bounds on the stabilizing number $sn(K)$ for $K= (4_1)_{(2,1)}$, namely $sn(\#_3(4_1)_{(2,1)})\ge 2$ and $sn(\#_6(4_1)_{(2,1)})\ge 3$, and reveal involutive obstructions to extending certain $\mathbb{Z}_2$-actions, suggesting strong cork phenomena surviving after stabilizations. The methods connect real Seiberg–Witten theory with cobordism constructions and branched-cover techniques to address smooth sliceness questions in 4-manifolds.

Abstract

We show that the 3-fold (resp. 6-fold) connected sum of the $(2,1)$-cable of the figure-eight knot cannot bound a smooth null-homologous disk in a punctured $S^2 \times S^2$ (resp. in a punctured $#_2 S^2 \times S^2$. This result is obtained using a real version of the $10/8$-inequality established by Konno, Miyazawa, and Taniguchi.

Theorems & Definitions (6)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• proof : Proof of \ref{['main thm']}
• Remark 2.1
• proof : Proof of \ref{['involution']}