On non almost-fibered knots

TL;DR

This paper demonstrates the existence of infinitely many hyperbolic genus one knots that are not almost-fibered, expanding the understanding of knot properties and classifications.

Contribution

It provides the first known examples of hyperbolic genus one knots that are not almost-fibered, challenging previous assumptions.

Findings

Existence of infinitely many non-almost-fibered hyperbolic genus one knots

Construction methods for such knots

Implications for knot classification theories

Abstract

An almost-fibered knot is a knot whose complement possesses a circular thin position in which there is one and only one weakly incompressible Seifert surface and one incompressible Seifert surface. Infinite examples of almost-fibered knots are known. In this article, we show the existence of infinitely many hyperbolic genus one knots that are not almost-fibered.