Related papers: An operad for splicing

We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate…

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…

We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of…

The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schw\"anzl and Vogt can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still…

For a knot $K$ in the 3-sphere and a simply connected closed 4-manifold $X$, we define the $X$-double slice genus of $K$, extending the notion from the case when $X$ is the 4-sphere. We show that for each integer $n$, there exists an…

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such…

We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a…

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.

Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot…

We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.

The join operad arises from the combinatorial study of the iterated join of simplices. We study a suitable simplicial version of this operad which includes the symmetries given by permutations of the factors of the join. From this…

This note describes how to construct toroidal polyhedra which are homotopic to a given type of knot and which admit an isohedral tiling of 3-space.

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of…

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying…

We determine the rational homology of the space of long knots in R^d for $d\geq4$. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E^1 page. As a corollary we get that the homology…

In this paper it is proved that each densely defined $J$-skew-symmetric operator (or each $J$-isometric operator with $\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a $J$-skew-self-adjoint (respectively $J$-unitary)…

We generalize the classical operad pair theory to a new model for $E_\infty$ ring spaces, which we call ring operad theory, and establish a connection with the classical operad pair theory, allowing the classical multiplicative infinite…

We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in…

We construct a simply connected $2-$complex $C$ embeddable in $3-$space such that for any embedding of $C$ in $\mathbb S^3$, any edge contraction forms a minor of the $2-$complex not embeddable in $3-$space. We achieve this by proving that…

Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic…