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

TL;DR

This paper proves that certain connected sums of the (2,1)-cable of the figure-eight knot cannot bound smooth null-homologous disks in specific 4-manifolds, using a real 10/8-inequality.

Contribution

It introduces a real version of the 10/8-inequality to study bounding properties of connected sums of knots in 4-manifolds.

Findings

3-fold connected sum cannot bound a null-homologous disk in punctured S^2 x S^2

6-fold connected sum cannot bound a null-homologous disk in punctured #_2 S^2 x S^2

Uses a novel real 10/8-inequality approach

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.