Casson Invariant Research Articles

Chirally Cosmetic Surgeries and Casson Invariants

We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the ${\rm SL}(2,\mathbb{C})$ version of Casson invariants. As applications, we give a complete classification of chirally cosmetic surgeries on alternating knots of genus one.

FORMULAS FOR THE CASSON INVARIANT OF CERTAIN INTEGRAL HOMOLOGY 3-SPHERES

In this paper, we introduce a representation of knots and links in S3 by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S3 by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

On the Casson Invariant Conjecture of Neumann–Wahl

In this article we prove the Casson Invariant Conjecture of Neumann–Wahl for splice type surface singularities. Namely, for such an isolated complete intersection we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber, provided that the link is an integral homology sphere.

Complex surface singularities with integral homology sphere links

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

Casson invariant of knots associated with divides

Casson invariant of knots associated with divides

Mapping class group and the Casson invariant

En utilisant une nouvelle definition des second et troisieme homomorphismes de Johnson, on simplifie et ou generalise les resultats de Morita sur l'invariant de Casson des spheres d'homologie definies par scindement de Heegard. En particulier, on calcule l'invariant de Casson des spheres d'homologie obtenues en recollant deux corps d'anses par un homeomorphisme du groupe de Torelli.

STABLE RANK-2 BUNDLES ON CALABI–YAU MANIFOLDS

In this paper, we apply the technique of chamber structures of stability polarizations to construct the full moduli space of rank-2 stable sheaves with certain Chern classes on Calabi–Yau manifolds which are anti-canonical divisor of ℙ1×ℙn or a double cover of ℙ1×ℙn. These moduli spaces are isomorphic to projective spaces. As an application, we compute the holomorphic Casson invariants defined by Donaldson and Thomas.

Equivariant Casson Invariants and Links of Singularities

This article is a survey of the authors' works over the last few years in trying to understand better various problems about the Casson invariant and Floer homology of links of singularities and, more generally, of graph homology spheres.

Seiberg–Witten and Casson–Walker Invariants for Rational Homology 3-Spheres

We consider a modified version of the Seiberg–Witten invariants for rational homology 3-spheres, obtained by adding to the original invariants a correction term which is a combination of η-invariants. We show that these modified invariants are topological invariants. We prove that an averaged version of these modified invariants equals the Casson–Walker invariant. In particular, this result proves an averaged version of a conjecture of Ozsvath and Szabo on the equivalence between their θ invariant and the Seiberg–Witten invariant of rational homology 3-spheres.

A Dehn surgery description of regular finite cyclic covering spaces of rational homology spheres

We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyond those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.

Lattice Points Inside Rational Simplices and the Casson Invariant of Brieskorn Spheres

We express the number of lattice points inside certain simplices with vertices in Q3 or Q4 in terms of Dedekind–Rademacher sums. This leads to an elementary proof of a formula relating the Euler characteristic of the Seiberg–Witten-Floer homology of a Brieskorn Z-homology sphere to the Casson invariant.

Equivariant Casson invariant for knots and the Neumann-Wahl formula

In 1985, Casson introduced a new 3-manifold invariant for integer homology spheres, via S'/7(2)-representation spaces of their fundamental groups. This invariant has solved some difficult problems in 3-manifold Topology and was found to have a relation to Gauge Theory, by work of Taubes in [17]. More recently, there have been various generalizations of Casson's invariant, for example Walker's extention to rational homology spheres. But more importantly for this article, in [3], Cappell, Lee and Miller have defined an equivariant Casson invariant, λ( /)(Ύ), for closed 3-manifolds Y with a semi-free Zn-action, where n is an odd prime and k an integer such that 0 < k < (n — l)/2. A particularly important case is that of n-fold cyclic branched coverings of S along a knot K, denoted Vn(K). In that case, the equivariant Casson invariants may be related to equivariant knot signatures of K, by developping a surgery formula for the invariants as done in [3]. In parallel, the author and B. Steer developped in [6] a Floer Homology for knots in 5, a knot invariant consisting of four abelian groups, denoted HF^(S^K), depending on a parameter 0 < a < 1/2. The Euler characteristic of this Floer Homology was related to equivariant knot signatures of K, via work of Herald in [9]. In fact, when α = k/n for n an odd prime, we have χ(HF^\S,K)) = λ<* ' ln\Vn(K}}, so that the Floer Homology for knots may be seen as a generalisation of the equivariant Casson invariant of cyclic branched covers of S along knots. In this article, we combine both points of view to define an equivariant Casson invariant for 3-manifolds which arise as 4-fold cyclic branched covers of S along a knot K. In the case of cyclic branched covers with an even branching index, the construction in [3] does not apply. The original motivation for this work was therefore to complement the work done in [3]. The resulting invariant brings new light to the Neumann-Wahl formula, relating Casson's invariant of a 3-dimensional link of complex singularity to the signature of its Milnor fibre, as explained in Section 4, where the equivariant Casson invariant is expressed in terms of the Jones polynomial of K, using a result of Mullins. The article ends with a generalization to other cyclic branched coverings of S with

A Geometric Proof of the Fintushel–Stern Formula

The Fintushel–Stern formula asserts that the Casson invariant of a Brieskorn homology sphere Σ(p, q, r) equals 1/8 the signature of its Milnor fiber. We give a geometric proof of this formula, as opposite to computational methods used in the original proof. The formula is also refined to relate equivariant Casson invariants to equivariant signatures.

Casson-Walker Invariant of Seifert Fibered Rational Homology Spheres as Quantum SO(3)-Invariant

In this paper, we shall give a general formula for the quantum SO(3)-invariant of Seifert fibered rational homology spheres over a 2-sphere, based on the linear skein theory. We shall derive the Casson-Walker invariant from the quantum SO(3)-invariant. Also, we shall give a formula for the Ohtsuki second invariant.

Unitary representations of Brieskorn spheres

In this article, we commence an investigation of the SU(N) representation space of Seifert fibered homology spheres Σ(a1, . . . , an). Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of Σ(a1, . . . , an) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian group. The latter representation spaces can be studied using the general correspondence between representations of Fuchsian groups and the moduli of parabolic bundles given by Mehta and Seshadri. For example, the inductive procedure of Atiyah-Bott-Nitsure determines the cohomology of this moduli space and it follows that the odd dimensional cohomology groups of any component of irreducible SU(N) representations of Σ(a1, . . . , an) vanish. In particular, any irreducible component of the SU(3) representation space of a Brieskorn spheres Σ(p, q, r) is either a point or a two sphere. By repeated application of the inductive procedure, the precise number of points and two spheres in this representation space is determined. Specific results for the Brieskorn spheres with p = 2 are given, where the representation space is a collection of points. In the last section, the SU(N) spectral flow of irreducible representations of Seifert fibered homology spheres is shown to be even. This gives a calculation of the leading term in a gauge-theoretic definition of the generalized Casson invariants.

Topological quantum field theories, moduli spaces, and flat gauge connections.

We show how to construct a topological quantum field theory which corresponds to a given moduli space. We apply this method to the case of flat gauge connections defined over a Riemann surface and discuss its relations with the Chern-Simons theory and conformal field theory. Geometrical properties are invoked to prove that the observables of those theories are not trivial. The case of the SO(2,1) group is separately discussed. A topological field theory is linked to the moduli space of self-dual'' connections over Riemann surfaces. Another relation between the Chern-Simons theory and topological quantum field theory in three dimensions is established. We present the theory which corresponds to three-dimensional gravity. Expressions for the Casson invariants are given. Possible generalizations are briefly discussed.