Exact triangles in monopole homology and the Casson-Walker invariant

通用不变量、康威多项式和卡森-沃克-莱斯科普不变量

Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant

本文通过图示和组合方法给出了闭3流形卡森-沃克-莱斯科普不变量的手术公式，揭示了康威多项式系数与 Kontsevich 积分系数的关系。

仅 cosmetic 手术与 Casson-Walker-Lescop 不变量

Purely cosmetic surgeries and Casson--Walker--Lescop invariants

该研究证明了在有理同调球面中，无向同调结至多有两个整数纯 cosmetic 手术，并探讨了相关 3-流形中结的同胚性质。

Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant III

For a knot $K$ in a homology $3$-sphere $Σ$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a Seifert fibered space with $N\ge 3$ singular fibers, then $N\ge 4$ if and only if the first homology of the universal abelian covering of $X$ is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of $K$, if $M$ is a Seifert fibered space, then $q=\pm 1$ (i.e.\ integral surgery).

The Casson-Walker-Lescop Invariant and Link Invariants

Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a coefficient of the Conway polynomial. Finally we compute Lescop's invariant of several lens spaces and deduce some values for Dedekind sums.

The theta divisor and the Casson-Walker invariant

We use Heegaard decompositions and the theta divisor on a Riemannian surface to define a three-manifold invariant for rational homology three-spheres. This invariant is defined on the set of $Spin^c$ structures $$ {\hat θ}\colon Spin^c(Y)\longrightarrow Q. $$ In the first part of the paper, we give the definition of the invariant (which builds on the theory developed in the article, "The theta divisor and three-manifold invariants"). In the second part, we establish a relationship between this invariant and the Casson-Walker invariant.

A vanishing result for a Casson-type instanton invariant

Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by a count of a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of PU(2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

The SU(3) Casson Invariant for Integral Homology 3-Spheres

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres.

Seiberg-Witten invariants of rational homology spheres

We prove that the Seiberg-Witten invariants of a rational homology sphere are determined in a very explicit fashion by the Casson-Walker invariant and the Reidemeister torsion

Splitting formulae for the Kontsevich-Kuperberg-Thurston invariant of rational homology 3-spheres

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behaviour of Z under rational homology handlebodies replacements. The explicit formulae that we present generalize a sum formula obtained by the author for the Casson-Walker invariant in 1994. They allow us to identify the degree one term of Z with the Walker invariant for rational hom…

Seiberg-Witten invariants and surface singularities III. Splicings and cyclic covers

We verify the conjecture formulated in math.AG/0111298 for suspension singularities of type $g(x,y,z)= f(x,y)+z^n$, where $f$ is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg-Witten invariant of the link $M$ of $g$, associated with the canonical $spin^c$ structure, equals $-σ(F)/8$, where $σ(F)$ is the signature of the Milnor fiber of $g$. In order to do this, we prove general splicing formulae for the Casson-Walker invariant and for the sign refined Reidemeister-Turaev torsion (in particular, for the modified Seiberg-Witten invariant too). These pr…

3-manifold topology and the Donaldson-Witten partition function

We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If $b_1(Y)>1$ we show that the partition function reduces to the Casson-Walker-Lescop invariant of $Y$, as expected on formal grounds. In the case $b_1(Y)=1$ there is a correction. Consequently, in the …

A splicing formula for the LMO invariant

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's fo…

Heegaard 图中 Theta-不变量的组合定义

A combinatorial definition of the Theta-invariant from Heegaard diagrams

本文基于Heegaard图在不依赖配置空间的情况下证明了$Θ$-不变量的组合公式定义了配对$(M,X)$的不变量，并通过手术公式证明该不变量等于$6λ(M)$与$p_1(X)/4$之和。

从图构型出发的链环与三维流形不变量

Invariants of links and 3-manifolds from graph configurations

本书定义并研究有理同调三维球面中带边链环的不变量Z，推广其为缠结范畴函子，计算迭代导数，结合他人结果表明其与LMO不变量信息等价且一度部分为Casson-Walker不变量。