Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Braun, Gábor; Némethi, András

doi:10.1515/crelle.2010.007

Cited by 22 publications

(77 citation statements)

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“…In the meantime, Okuma resp. Némethi and Okuma in [24,13,14] (see also Braun and Némethi [2]) have used this to compute the geometric genus p g of any splice quotient, and to prove for splice quotients the Casson invariant conjecture [17] for singularities with ZHS links (in which case D = {1} so V = X), as well as the Némethi-Nicolaescu extension [12] of the Casson Invariant Conjecture to singularities with QHS links.…”

Section: Our Main Results Ismentioning

confidence: 99%

The end curve theorem for normal complex surface singularities

Neumann

Wahl

2010

J. Eur. Math. Soc.

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Abstract. We prove the "End Curve Theorem," which states that a normal surface singularity (X, o) with rational homology sphere link is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.An "end curve function" is an analytic function (X, o) → (C, 0) whose zero set intersects in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.A "splice quotient singularity" (X, o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in C t , where t is the number of leaves in the resolution graph for (X, o), together with an explicit description of the covering transformation group.Among the immediate consequences of the End Curve Theorem are the previously known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).Keywords. Surface singularity, splice quotient singularity, rational homology sphere, complete intersection singularity, abelian cover, numerical semigroup, monomial curve, linking pairing We consider normal surface singularities whose links are rational homology spheres (QHS for short). The QHS condition is equivalent to the condition that the resolution graph of a minimal good resolution be a rational tree, i.e., is a tree and all exceptional curves are genus zero.Among singularities with QHS links, splice quotient singularities are a broad generalization of weighted homogeneous singularities. We recall their definition briefly here and in more detail in Section 1. Full details can be found in [20].Recall first that the topology of a normal complex surface singularity is determined by and determines the minimal resolution graph . Let t be the number of leaves of . For i = 1, . . . , t, we associate the coordinate function x i of C t to the i-th leaf. This leads to a natural action of the "discriminant group" D = H 1 ( ) by diagonal matrices on C t (see Section 1).

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Section: Our Main Results Ismentioning

confidence: 99%

The end curve theorem for normal complex surface singularities

Neumann

Wahl

2010

J. Eur. Math. Soc.

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“…The conjectured identities were verified for important families of singularities, e.g. for splice quotient singularities [23,2]. The connections continue at cohomology level as well.…”

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confidence: 67%

“…The lattice cohomology has subtle connections with a certain multivariable Poincaré series (defined combinatorially from the graph, which resonates and sometimes equals the multivariable Poincaré series associated with the divisorial filtration indexed by all the divisors in the resolution, provided by certain analytic realizations) [15,17,19]. For example, the Seiberg-Witten invariant appears as the 'periodic constant' of this series [17,2,23]. (We review these facts in Section 5.)…”

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confidence: 99%

Reduction Theorem for Lattice Cohomology

Tamás¹,

Némethi²

2014

International Mathematics Research Notices

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Abstract. The lattice cohomology of a plumbed 3-manifold M associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of M and in the comparison of the topological properties with analytic ones, whenever M is realized as complex analytic singularity link. By definition, its computation is based on the (Riemann-Roch) weights of the lattice points of Z s , where s is the number of vertices of the plumbing graph. The present article reduces the rank of this lattice to the number of 'bad' vertices of the graph. (Usually the geometry/topology of M is codified exactly by these 'bad' vertices via surgery or other constructions. Their number measures how far is the plumbing graph from a rational one, or, how far is M from an L-space.)The effect of the reduction appears also at the level of certain multivariable (topological Poincaré) series as well. Since from these series one can also read the Seiberg-Witten invariants, the Reduction Theorem provides new formulae for these invariants too.The reduction also implies the vanishing H q = 0 of the lattice cohomology for q ≥ ν, where ν is the number of 'bad' vertices. (This bound is sharp.)

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“…Consider the plumbing graph of a superisolated singularity corresponding to a curve of degree d = 8 with three singular points whose knots K 1 , K 2 , K 3 are the torus knots of type (6, 7), (2, 9), (2,5), respectively.…”

Section: Examples and Applicationsmentioning

confidence: 99%

Seiberg–Witten Invariant of the Universal Abelian Cover of $${S_{-p/q}^{3}(K)}$$

Bodnár

Némethi

2017

Singularities and Computer Algebra

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Abstract. We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singularities in several steps of the proof. This topological covering additivity property can be compared with certain analytic properties of normal surface singularities, especially with functorial behaviour of the (equivariant) geometric genus of singularities. We present several examples in order to find the validity limits of the proved property, one of them shows that the covering additivity property is not true for negative definite plumbed 3-manifolds in general.

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Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds

Cited by 22 publications

References 21 publications

The end curve theorem for normal complex surface singularities

The end curve theorem for normal complex surface singularities

Reduction Theorem for Lattice Cohomology

Seiberg–Witten Invariant of the Universal Abelian Cover of $${S_{-p/q}^{3}(K)}$$

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