2
$\begingroup$

In $S^3$ we have a well defined connected sum operation of knots, $K_1 \# K_2$ and a band connected sum $K_1 \#_b K_2$, which is not well defined and depends on the position of the band $b$. What are the conditions on $b$, so that the band connected sum would be well defined?

How is the story in a general 3-manifold $M$? If $K_1$ is a general knot in $M$ and $K_2$ is a local knot (inside a 3-ball), then $K_1 \# K_2$ is well defined, what conditions would $b$ have to satisfy so that $K_1 \#_b K_2$ would be well defined?

$\endgroup$

1 Answer 1

0
$\begingroup$

A probably unhelpful partial answer is that if for $K=K_1\#_b K_2$ there is a sphere $\Sigma$ such that $\Sigma\cap K$ is two points, one on each strand of the band, then $K=K_1\# K_2$. This is because you can do something like the argument for the lightbulb theorem using $\Sigma$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .