2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73254We show that the free energy of the topological string admits a certain pole structure by using the operator formalism. Combined with the results of Peng that proved the integrality, this gives a combinatoric proof of the Gopakumar-Vafa conjecture.37 pages; v2: materials reorganized and references added, 46 pages; v3:title changed, to appear in Publ. Res. Inst. Math. Sci. KyotoAlgebraic GeometryCombinatorics14N35; 05E05text