Toroidal Seifert fibered surgeries on alternating knots

TL;DR

This paper classifies all alternating knots that admit toroidal Seifert fibered surgeries, identifying specific knots and surgery slopes where these conditions occur, thus completing the understanding of such surgeries in this class.

Contribution

It provides a complete classification of toroidal Seifert fibered surgeries on alternating knots, detailing the specific knots and slopes involved.

Findings

Trefoil knot with zero surgery slope admits such a surgery.

Connected sums of (2,p)- and (2,q)-torus knots with slope 2(p+q) admit such surgeries.

No other alternating knots admit toroidal Seifert fibered surgeries.

Abstract

We give a complete classification of toroidal Seifert fibered surgeries on alternating knots. Precisely, we show that if an alternating knot admits a toroidal Seifert fibered surgery, then the knot is either the trefoil knot and the surgery slope is zero, or the connected sum of a (2,p)-torus knot and a (2,q)-torus knot and the surgery slope is 2(p+q) with |p|, |q| at least three.