The Petersen graph is nonplanar by an explicit subdivision of $K_{3,3}$ after deleting one vertex

Delete the vertex $12$. Take the branch classes $A=\{13,14,15\}$ and $B=\{23,24,25\}$. The six direct branch connections are $13$ to $24,25$, $14$ to $23,25$, and $15$ to $23,24$. The remaining three are the paths $13-45-23$, $14-35-24$, and $15-34-25$.

Every consecutive pair in the displayed list has disjoint labels, so it is an edge by [F1]. The internal vertices $45,35,34$ are distinct and are not branch vertices; all other listed connections are single edges. Thus the nine paths are internally disjoint and form a subdivision, in the sense of Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors, of $K_{3,3}$ with branch classes $A$ and $B$. This subdivision lies in a subgraph of the Petersen graph, so [L1] proves that the Petersen graph is nonplanar.