Beijing Institute of Mathematical Sciences and Applications

Vassily Manturov , Shiquan Ren , Zhe Yan Wan

Arkadiy Skopenkov

Wednesday, August 21, 2024 3:30 PM - 5:00 PM

A3-2-301

Zoom 831 5020 0580 (141592)

Let $f:S^q\bigsqcup S^q\to S^m$ be an (ordered oriented) 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 $\sigma f:S^q\bigsqcup S^q\to S^m$ 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 $S^q\bigsqcup S^q\to S^m$ obtained by embedded connected sum of the last two components of a link $g:S_1^q\bigsqcup S_2^q\bigsqcup S_3^q\to S^m$ 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 $S^6$.