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Séminaire Lotharingien de Combinatoire, B84c (2022), 21 pp.

# Jan Geuenich, Daniel Labardini-Fragoso and José Luis Miranda-Olvera

# Quivers with Potentials Associated to Triangulations of Closed
Surfaces with At Most Two Punctures

**Abstract.**
We tackle the classification problem of non-degenerate potentials for
quivers arising from triangulations of surfaces in the cases left open
by
Geiss, Labardini-Fragoso and Schröer. Namely, for once-punctured
closed surfaces of positive genus, we show that the quiver of any
triangulation admits infinitely many non-degenerate potentials that
are pairwise not weakly right-equivalent; we do so by showing that the
potentials obtained by adding the 3-cycles coming from triangles and a
fixed power of the cycle surrounding the puncture are well behaved
under flips and QP-mutations. For twice-punctured closed surfaces of
positive genus, we prove that the quiver of any triangulation admits
exactly one non-degenerate potential up to weak right-equivalence,
thus confirming the veracity of a conjecture of the aforementioned
authors.

Received: January 31, 2021.
Accepted: February 8, 2022.
Final Version: February 8, 2022.

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