Dehn surgeries on knots in product manifolds

TL;DR

This paper characterizes knots in product sutured manifolds that admit surgeries resulting in the same manifold, identifying them as 0- or 1-crossing knots using sutured manifold theory techniques.

Contribution

It provides a classification of knots based on their surgery outcomes in product sutured manifolds, linking knot complexity to surgery invariance.

Findings

Knots with surgeries yielding the same product manifold are 0- or 1-crossing knots.

Uses sutured manifold theory to establish the classification.

Offers new insights into the relationship between knot complexity and manifold invariance.

Abstract

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The proof uses techniques from sutured manifold theory.