Non-integer surgery and branched double covers of alternating knots

TL;DR

This paper demonstrates that non-integer surgeries on knots in the 3-sphere producing branched double covers of alternating links can be understood through rational tangle replacements in their diagrams, linking surgery theory with diagrammatic modifications.

Contribution

It establishes a direct connection between non-integer surgeries yielding alternating link covers and rational tangle replacements in their diagrams, providing a new perspective on the structure of such surgeries.

Findings

Non-integer surgeries correspond to rational tangle replacements.

Branched double covers of alternating links can be characterized by diagrammatic modifications.

The work links surgery theory with combinatorial diagram transformations.

Abstract

We show that if the branched double cover of an alternating link arises as $p/q \in \mathbb{Q} \setminus \mathbb{Z}$ surgery on a knot in $S^3$, then this is exhibited by a rational tangle replacement in an alternating diagram.