Allowable Closed Surface Evaluations From Topological Quantum Field Theory

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Every 2D TQFT evaluates closed 2D surfaces to an element of the base field. These evaluations appear as a fundamental part of the skein relations induced by the TQFT, dating back to Bar-Natan. We definitively classify what sets of closed surface evaluations may arise from a 2D TQFT. Our answer reveals a deep relationship between achievable evaluations and the theory of symmetric polynomials. We then apply our results to precisely determine which 2D TQFTs have an associated Frobenius algebra that may be realized as the algebra of all surfaces with boundary S-1, modulo surfaces that are "evaluated similarly" by the TQFT.