The Crossing Number as a Gromov-Witten Invariant: A Geometricization of Graph Embeddings

<p dir="ltr">This paper builds a deep and unexpected bridge between the combinatorial world of graph theory and the geometric world of algebraic geometry. We show that a fundamental and notoriously difficult combinatorial invariant—the crossing number of a graph, which counts the minimum number of times its edges must cross when drawn in the plane—can be understood as a specific instance of a Gromov-Witten invariant, a sophisticated tool from enumerative geometry.</p><p dir="ltr">To achieve this, we introduce a new geometric object: a specially crafted algebraic surface built from the data of the graph. We then define "algebraic drawings" of the graph as certain geometric maps from this surface to the projective plane. Our main theorem proves that the classical crossing number is equal to a minimized intersection count calculated from these algebraic drawings.</p><p dir="ltr">This minimum count is exactly computed by a new, graph-defined Gromov-Witten invariant. This correspondence recasts the elusive crossing number problem into a concrete problem of counting curves in algebraic geometry. We demonstrate the power of this new framework by using techniques from equivariant geometry to perform explicit calculations for small, complex graphs, successfully recovering the long-known but difficult results for the complete graphs with 5 and 6 vertices.</p><p dir="ltr">Finally, this new perspective allows us to derive fresh conjectures about the crossing numbers of graph products, stemming directly from the structural properties of Gromov-Witten theory, and reveals new combinatorial structures within classical moduli spaces.</p>