Related papers: A Note on Knot Concordance
Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…
We propose and analyze a structure with which to organize the difference between a knot in the 3-sphere bounding a topologically embedded 2-disk in the 4-ball and it bounding a smoothly embedded disk. The n-solvable filtration of the…
We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the…
We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many…
We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish…
We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a…
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…
Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in $S^3$…
We develop monopole and instanton Floer homology groups for balanced sutured manifolds. Applications include a new proof of Property P for knots.
Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds…
We show that every $PD_3$-complex $P$ bounds a $PD_4$-pair $(Z,P)$. If $P$ is orientable we may assume that $\pi_1(Z)=1$. We show also that if $P$ has a manifold 1-skeleton then it is homotopy equivalent to a closed 3-manifold, and that if…
Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but…
In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard Diagrams of 3-manifolds. For a knot inside S3, we further develop an algorithm…
We give a short proof of Bing's characterization of $S^3$: a compact, connected 3-manifold $M$ is $S^3$ if and only if every knot in $M$ is isotopic into a ball.
If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$…
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of…
In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
An oriented compact closed manifold is called inflexible if the set of mapping degrees ranging over all continuous self-maps is finite. Inflexible manifolds have become of importance in the theory of functorial semi-norms on homology.…
We prove a recent conjecture of Manolescu-Willis which states that the $s$-invariant of a knot in $\mathbb{RP}^3$ (as defined by them) gives a lower bound on its null-homologous slice genus in the unit disk bundle of $TS^2$. We also…