Related papers: Monopoles and transverse knots
We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the…
We consider a connected negative definite plumbing graph, and we assume that the associated plumbed 3-manifold is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of this manifold. The…
The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their…
Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on $\mathbb{CP}^2$ and infinitely many exotic smooth…
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the spaces of holomorphic sections of a prequantizing line bundle over compact K\"ahler manifolds under deformations of the complex structure. We show that the…
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology…
We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, a reflection functor and a transpose defined by certain bilinear forms. These…
Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…
We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozv\'ath-Szab\'o. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines…
Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational…
This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently…
Let $K$ be a knot with an unknotting tunnel $\gamma$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $\rho = p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, \gamma)$. The invariant $\rho$ has…
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
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.…
We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families…
[Original abstract (1992):] The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a…
Suppose that $X$ is a torus bundle over a closed surface with homologically essential fibers. Let $X_K$ be the manifold obtained by Fintushel--Stern knot surgery on a fiber using a knot $K\subset S^3$. We prove that $X_K$ has a symplectic…
Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables…
We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a…