Dehn surgery on complicated fibered knots in the 3-sphere

TL;DR

This paper proves that fibered knots in the 3-sphere with sufficiently complicated monodromies cannot produce lens spaces through Dehn surgery, linking monodromy complexity to surgery outcomes.

Contribution

It establishes a new connection between the complexity of monodromy in fibered knots and the impossibility of lens space surgeries for highly complicated cases.

Findings

Complicated monodromy prevents lens space surgeries on fibered knots.

Fibered knots with lens space surgeries have relatively simple monodromy.

The result complements Yi Ni's work on fibered knots and lens space surgeries.

Abstract

Let K be a fibered knot in the 3-sphere. We show that if the monodromy of K is sufficiently complicated, then Dehn surgery on K cannot yield a lens space. Work of Yi Ni shows that if K has a lens space surgery then it is fibered. Combining this with our result we see that if K has a lens space surgery then it is fibered and the monodromy is relatively simple.