The band connected sum and the second Kirby move for higher-dimensional links

TL;DR

This paper investigates how connected sums of components in higher-dimensional links affect their isotopy classes, providing answers for specific dimensions and applying results to classify linked 3-manifolds in S^6.

Contribution

It introduces new insights into the dependence of link isotopy classes on connected sum operations in higher dimensions, specifically for the case q=4k-1 and m=6k.

Findings

Derived formulas for the effect of connected sums on link isotopy classes

Applied results to classify linked 3-manifolds in S^6

Extended understanding of the second Kirby move in higher dimensions

Abstract

Let $f:S^q\sqcup S^q\to S^m$ be a link (i.e. an embedding). How does (the isotopy class of) the knot $S^q\to S^m$ obtained by embedded connected sum of the components of $f$ depend on $f$? Define a link $\sigma f:S^q\sqcup S^q\to S^m$ as follows. The first component of $\sigma f$ is the `standardly shifted' first component of $f$. The second component of $\sigma f$ is the embedded connected sum of the components of $f$. How does (the isotopy class of) $\sigma f$ depend on $f$? How does (the isotopy class of) the link $S^q\sqcup S^q\to S^m$ obtained by embedded connected sum of the last two components of a link $g:S^q_1\sqcup S^q_2\sqcup S^q_3\to S^m$ depend on $g$? We give the answers for the `first non-trivial case' $q=4k-1$ and $m=6k$. The first answer was used by S. Avvakumov for classification of linked 3-manifolds in $S^6$.