Genera of knots in the complex projective plane

TL;DR

This paper systematically computes the complex projective plane genus for all prime knots up to 8 crossings, using bounds from band surgery and homological obstructions, revealing differences in genus between knots and their mirrors.

Contribution

It provides the first comprehensive determination of the $ ext{CP}^2$-genus for prime knots up to 8 crossings, introducing new obstruction techniques and identifying knots with asymmetric genus properties.

Findings

Determined $ ext{CP}^2$-genus for all but 2 knots up to 7 crossings.

Determined $ ext{CP}^2$-genus for all but 9 knots up to 8 crossings.

Identified an infinite family of knots with genus differing from their mirrors.

Abstract

Our goal is to systematically compute the $\mathbb{C}P^2$-genus of all prime knots up to 8-crossings. We obtain upper bounds on the $\mathbb{C}P^2$-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to $\mathbb{C}P^2$. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the $\mathbb{C}P^2$-genus of all but 2 of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the $\mathbb{C}P^2$-genus of all but 9 of these knots. Where the $\mathbb{C}P^2$-genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the $\mathbb{C}P^2$-genus of each knot differs from that of it's mirror.