Surgery formulae for the Seiberg–Witten invariant of plumbed 3-manifolds
Tóm tắt
Assume that $$M({{\mathcal {T}}})$$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $$\mathcal {T}$$. We consider the combinatorial multivariable Poincaré series associated with $$\mathcal {T}$$ and its counting functions, which encode rich topological information. Using the ‘periodic constant’ of the series (with reduced variables associated with an arbitrary subset $${{\mathcal {I}}}$$ of the set of vertices) we prove surgery formulae for the normalized Seiberg–Witten invariants: the periodic constant associated with $${{\mathcal {I}}}$$ appears as the difference of the Seiberg–Witten invariants of $$M({{\mathcal {T}}})$$ and $$M({{\mathcal {T}}}{\setminus }{{\mathcal {I}}})$$ for any $${{\mathcal {I}}}$$.
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