Non-normal affine monoids, modules and Poincaré series of plumbed 3-manifolds

Ehrhart theory of polytopes and Seiberg-Witten invariants of plumbed 3-manifolds

Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytopes from the plumbing graphs together with an action of the first homology of M, and we develop Ehrhart theory for them. At an intermediate level we define the `periodic constant' of multivariable series and establish their properties. In this way, one identifies the Seiberg-Witten invariant of a plumbed 3-manifo…

Cobordism category of plumbed 3-manifolds and intersection product structures

In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we also consider the homology cobordism monoid, and give a necessary condition using w-invariants for the homology 3-spheres to belong to the inertia group associated to some homology 3-spheres.

Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds

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) we prove surgery formulae for the normalized Seiberg-Witten invariants: the periodic constant appears as the difference of the Seiberg-Witten invariants associated with $M(\mathcal{T})$ and $M(\mathcal{T}\setminus\mathcal{I})…

On Poincaré series associated with links of normal surface singularities

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seib…

积分同调球面的Poincaré级数的拼接公式

Splice formulae for Poincaré series of integral homology spheres

本文针对与负定拼接图相关的积分同调球面3-流形，给出其多变量Poincaré级数及其多项式部分的拼接公式，修正项为易计算的单变量Alexander多项式，从而可由较小图恢复全局不变量。

无穷多族量子模3-流形不变量

Infinite Families of Quantum Modular 3-Manifold Invariants

本文利用AJK理论扩展了3-流形的格上同调与BPS q-级数，首次为无穷多族负定plumbed 3-流形计算了AJK级数，并构造了可视为WRT不变量形变且具量子模性质的无穷多族不变量。