Related papers: A Dunfield--Gong 4-Sphere is Standard

We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures.…

An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.

We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice…

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from…

We show that if $K$ is a knot in $S^3$ and $\Sigma$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $\Sigma$ via an isotopy is $n$. We…

We call a non-trivial homology sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur or Po\'enaru manifold. In 1980, Kirby found the first example by proving that the boundary of the Ramanujam surface bounds a Mazur…

In $1961$, Mazur constructed a contractible, compact, smooth $4$-manifold with boundary which is not homeomorphic to the standard $4$-ball, using a $0$-handle, a $1$-handle and a $2$-handle. In this paper, for any integer $n\geq2,$ we…

Let $K, K'$ be ribbon knottings of $n$-spheres with $1$-handles in $S^{n+2}$, $n\geq 2$. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and…

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an…

In this paper, we generalize a result of Satoh to show that for any odd natural $n$, the connected sum of the $n$-twist spun sphere of a knot $K$ and an unknotted projective plane in the 4-sphere is equivalent to the same unknotted…

We prove a generalization of a result by Dong and Santos-Sturmfels about the homotopy type of the Alexander dual of balls and spheres. Our results involve NH-manifolds, which were recently introduced as the non-homogeneous (or non-pure)…

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even…

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…

We extend the infinite family of exotic embeddings $\mathbb{RP}^2 \hookrightarrow S^4$ constructed by Miyazawa to a strictly larger family of exotic embeddings, by showing that in place of the pretzel knot $P(-2, 3, 7)$, an infinite family…

Iwase and Matsumoto defined `pochette surgery' as a cut-and-paste on 4-manifolds along a 4-manifold homotopy equivalent to $S^2\vee S^1$. The first author in [10] studied infinitely many homotopy 4-spheres obtained by pochette surgery. In…

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a…

We prove that, for a closed oriented smooth spin 4-manifold $X$ with non-zero signature, the Dehn twist about a $(+2)$- or $(-2)$-sphere in $X$ is not homotopic to any finite order diffeomorphism. In particular, we negatively answer the…

The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower…

In the present study we consider knotted spheres in Euclidean $4$-space $ \mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $ \mathbb{E}^{4}$. Further, we obtained some results related with the…

Conjecturally, a knot is slice if and only if its positive Whitehead double is slice. We consider an analogue of this conjecture for slice disks in the four-ball: two slice disks of a knot are smoothly isotopic if and only if their positive…