arXiv research

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah-Hitchin-Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang-Mills moduli spaces are constructed as finite dimensional smooth manifolds over Cass…

The study distinguishes exotic R^4's using knot Floer homology and Casson handles.

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…

In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with …

We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson han…

Kirby diagrams for exotic R^4's constructed from specific knot complements.

New bounds and examples for sphere unknotting numbers.

New theorem connects handle-ribbon knots to slice derivatives.

Defines a new $SL_2(\mathbb{R})$-Casson invariant using Reidemeister torsions.

New exotic 4D spaces found using knot slicing techniques.

This paper proves a symplectic formula for SU(n) generalized Casson invariants.

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory.

We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.

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

We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.

Formula for Casson-Walker invariant; applies to knot complements and cosmetic crossing conjecture.

We prove the additivity of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ under fiber sum along embedded curves and embedded tori, which is the $4$-dimensional analogue of the additivity of the Casson invariant under connected-sum and splicing along knots.

We show that Scharlemann-Thompson untelescoping of Heegaard splittings is finer than Casson-Gordon's by giving concrete examples.

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing…

We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.

It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show …

I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson towe…

We compute the $SU(2)$ Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.

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…

In this paper, we extend the definition of the $SL_2(\Bbb C)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\widehat{A}$-polynomial of $K$. We prove a product formula for the $\widehat{A}$-polynomial of the connected sum $K_1 \# K_2$ of two knots in $S^3$…

We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combin…

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This pap…

Lower bound on stable 4-genus of knots using Casson-Gordon signatures.

The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing $n$-component links with a Dehn surgery realizing $\#^n(S^1\times S^2)$. We call such links $n$R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diff…

We prove that the ($τ$-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant introduced by Manolescu and the first author in [CM19] is generically independent of the parameter $τ$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked in [CM19]. Our arguments inv…

A three dimensional supergravity theory which generalizes the super IG theory of Witten and resembles the model discussed recently by Mann and Papadopoulos is displayed. The partition function is computed, and is shown to be a three-manifold invariant generalizing the Casson invariant.

New invariants for 3-manifolds derived from equivariant Cerf theory.

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson-Walker-Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points.

A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…

We investigate the behavior of the SL(2,C) Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler--Shalen seminorms, and we illustrate this approach by providing explicit computations …

In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group.

This is a research announcement on an alternative definition of the Casson invariants by means of virtual counting of the moduli space of irreducible representations of the fundamental group into $\SU(2)$. Along the way, by using derived differential geometry, we propose a general framework to obtain invariants from Ch…

We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge kno…

We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels, where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(…

The paper connects knot representations and spherical quandle colorings.

New formula for 3-manifold invariants using combinatorial methods.

The paper calculates a specific invariant for a manifold with a circle action.

The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional …

Study shows quasi-additivity of Δ-unknotting number for braids.

We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\T$, of the element of $\T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotica…

The study characterizes lamination limits and homeomorphisms in 3D handlebodies.