Immersed surfaces and Seifert fibered surgery on Montesinos knots

TL;DR

This paper investigates conditions under which Montesinos knots do not admit atoroidal Seifert fibered surgeries, using immersed surfaces to establish new restrictions based on the parameters of the knots.

Contribution

It introduces a novel approach employing immersed surfaces to analyze Seifert fibered surgeries on Montesinos knots, providing specific parameter constraints for non-existence.

Findings

No atoroidal Seifert fibered surgery if 1/(q1-1)+1/(q2-1)+1/(q3-1) ≤ 1

Provides a new method using immersed surfaces for studying knot surgeries

Establishes parameter bounds for Montesinos knots regarding Seifert fibered surgeries

Abstract

We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.