# On Genus g Orientable Crossing Numbers of Small Complete Graphs

## TL;DR

This paper reviews known results and presents new findings on genus g orientable crossing numbers of small complete graphs, including exact values and embeddings for K9 and K11.

## Contribution

It provides new exact crossing number values for K10 and K11 in genus 3 and 4, and identifies specific embeddings of K9 on genus 3 surfaces.

## Key findings

- cr3(K10)=3
- cr3(K11)≤14
- cr4(K11)=4

## Abstract

The current state of knowledge of genus g orientable crossing numbers of complete graphs through K11 is reviewed. It is shown that cr3(K10)=3, cr3(K11)<=14, and cr4(K11)=4. It is established with the aid of an algorithm that there are precisely two non-isomorphic embeddings of K9 with a hexagon with all its vertices distinct on a surface of genus 3.