Obstructing Sliceness in a Family of Montesinos Knots

TL;DR

This paper demonstrates that a specific family of Montesinos knots cannot be smoothly slice by applying gauge theory and Donaldson's theorem, extending previous methods used for simpler knots.

Contribution

It introduces a new application of gauge theoretical techniques to obstruct sliceness in a broad family of Montesinos knots, generalizing prior results for simpler knot types.

Findings

No knots in the specified family are smoothly slice under given conditions.

The techniques involve Donaldson's diagonalization theorem and properties of bounding plumbing 4-manifolds.

Some examples have signature zero and square determinant, illustrating the method's effectiveness.

Abstract

Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's diagonalization theorem and the fact that the 2-fold covers of Montisinos knots bound plumbing 4-manifolds, many of which are negative definite. Some of our examples include knots with signature 0 and square determinant.