Related papers: On closed oriented surfaces in the 3-sphere

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous…

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have…

In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of…

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a…

Let $N_k$ denote the closed non-orientable surface of genus $k$. In this paper we study the behaviour of the `square map' from the group of isometries of hyperbolic 3-space to the subgroup of orientation preserving isometries. We show that…

As a generalization of the classical knots, knotoids are equivalence classes of immersions of the oriented unit interval in a surface. In recent years, a variety of invariants of spherical and planar knotoids have been constructed as…

The paper considers the uniqueness question of factorization of a knotted handlebody in the $3$-sphere along decomposing $2$-spheres. We obtain a uniqueness result for factorization along decomposing $2$-spheres meeting the handlebody at…

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli…

Let f, g be two homotopic smooth embeddings of a closed surface in a closed oriented 5-dimensional manifold. We show that if f admits a common algebraic dual 3-sphere, or if the fundamental group of the ambient space is trivial, then f and…

Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. We prove that the number of minimal discs in $\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is finite and…

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots…

We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant and compute the…

This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a…

We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus $g$, the number of boundary components $b$, and the boundary slope $p/q$ are fundamental invariants of an essential surface. The \textit{realization…

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in $S^3$ that do not become isotopic when their interiors are pushed into $B^4$. In particular, we identify examples where the surfaces…

We fix a null-homologous, homotopically essential knot $J$ in a 3-manifold with PTFA fundamental group and study concordance of knots that are homotopic to $J$. We construct an infinite family of knots that are characteristic to $J$, and…

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections…

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a…

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and…