Constructing knots with low rational genera

TL;DR

This paper introduces a flexible method for constructing knots with low rational genus in 3-spheres and other 4-manifolds, revealing new phenomena in knot genus behavior across different homology balls.

Contribution

It provides the first examples of knots with genus differences in distinct homology balls and shows that every knot bounds a Möbius band in a rational homology ball.

Findings

Constructed knots with low genus in punctured open books.

First examples of genus variation in different homology balls.

Knots bounding Möbius bands in rational homology balls.

Abstract

We give a flexible construction for knots in the 3-sphere that bound surfaces of unexpectedly low genus in punctured open books on 3-manifolds. We use this construction to give the first examples of knots whose genus differs in different $\mathbb{Z}/2\mathbb{Z}$ homology balls. We also establish that every knot bounds a M{\"o}bius band in a rational homology ball, and that there are knots whose genus in $T^4$ and $B^4$ differ arbitrarily.