Surgery on links with unknotted components and three-manifolds

TL;DR

This paper proves that any closed three-manifold obtained by integral surgery on a knot can also be constructed from surgeries on a 3-component unlink, highlighting the versatility of unknotted links in three-manifold construction.

Contribution

It demonstrates that any three-manifold from knot surgery can be realized via surgeries on a 3-component unlink, revealing new insights into link surgeries and manifold representations.

Findings

Any closed three-manifold from knot surgery can be obtained from a 3-component unlink.

Multiple different surgeries on the same unlink can produce the same three-manifold.

Unknotted links are sufficient for representing all such three-manifolds.

Abstract

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M.