Non-simple genus minimizers in lens spaces

TL;DR

This paper constructs numerous non-simple knots that minimize the rational genus in lens spaces, challenging the prior assumption that only simple knots serve this role, thus providing new insights into knot theory and lens space surgeries.

Contribution

It introduces a method to construct many non-simple genus minimizers, expanding the understanding of minimal genus knots beyond simple knots in lens spaces.

Findings

Many non-simple genus minimizers are constructed.

This challenges the belief that only simple knots minimize rational genus.

The results impact the understanding of the Berge conjecture and lens space surgeries.

Abstract

Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non-simple genus minimizers. This negatively answers a question of Rasmussen.