A good presentation of (-2,3,2s+1)-type Pretzel knot group and R-covered foliation

TL;DR

This paper investigates the conditions under which certain Pretzel knot surgeries do not admit R-covered foliations, extending previous results to a broader class of knots.

Contribution

It generalizes prior work by proving that for (-2,3,2s+1)-Pretzel knots with specific surgery parameters, the resulting manifolds lack R-covered foliations.

Findings

For q > 0, p/q >= 4s+7, and p odd, the surgery manifold does not contain an R-covered foliation.

Extends previous results from the (-2,3,7) Pretzel knot to a wider family of Pretzel knots.

Provides conditions that prevent the existence of certain foliations in knot surgery manifolds.

Abstract

Let K_s be a (-2,3,2s+1)-type Pretzel knot (s >= 3) and E(K_s)(p/q) be a closed manifold obtained by Dehn surgery along K_s with a slope p/q. We prove that if q>0, p/q >= 4s+7 and p is odd, then E(K_s)(p/q) cannot contain an R-covered foliation. This result is an extended theorem of a part of works of Jinha Jun for (-2,3,7)-Pretzel knot.