Every weakly negative definite plumbing tree can be reduced to a negative definite one by a finite sequence of Neumann moves, with an explicit algorithm combining plumbing calculus and a diagonalization procedure.
Root lattices and invariant series for plumbed 3-manifolds
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abstract
We study formal series which are invariants of plumbed 3-manifolds twisted by root lattices. These series extend the BPS $q$-series $\widehat{Z}(q)$ recently defined in Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and further refined in Ri. We show that the series $\widehat{Z}(q)$ is unique in an appropriate sense and decomposes as the average of related series which are themselves invariant under the five Neumann moves amongst plumbing trees. Explicit computations are presented in the case of Brieskorn spheres and a non-Seifert manifold.
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math.GT 1years
2026 1verdicts
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Plumbed 3-Manifolds and Neumann Moves
Every weakly negative definite plumbing tree can be reduced to a negative definite one by a finite sequence of Neumann moves, with an explicit algorithm combining plumbing calculus and a diagonalization procedure.