arXiv research

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

4-ball can be tiled with knotted surfaces.

New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.

Study shows surgeries on certain knots bound rational homology 4-balls.

Same-genus Seifert surfaces for non-split alternating links are smoothly isotopic.

We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which …

Paper tackles which 3-spheres bound contractible 4-manifolds or homology 4-balls.

Classifies knots that bound equivariant surfaces with free symmetries.

The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.

New non-isotopic Seifert surfaces found in 4-ball.

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

We found an infinite family of counterexamples to Batson's conjecture.

New Seifert surfaces in 4-ball differ even when pushed in.

A link in the 3-sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the 4-ball. More generally, given a 4-manifold M with a distinguished circle in its boundary, a link in the 3-sphere is called M-slice if its components bound in the 4-ball disjoint embedded copies of M. A 4-ma…

Counterexamples found for knot conjectures.

In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is…

Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$ analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let $K$ be a smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a sharp lower bound for the 4…

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

New surfaces in 4-ball constructed from knits, described by charts.

Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way,…

New spanning tree model connects knot homology, s-invariant, and exotic discs.

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…

New 3-manifolds bound rational 4-balls through specific operations.

The paper quantifies how much of a 4-ball must be removed to squeeze into a cylinder, proving a lower bound on the Minkowski dimension.

The paper presents fundamental groups of complements of shadows in 4-balls.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.…

The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot …

New knots not slice in rational 4-balls found.

We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in $\mathbb{C}^2$ with the unit 4-ball from which a 4-ball of smaller radius is…

Lisa Piccirillo solved the mystery of the Conway knot's sliceness.

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, al…

A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-…

New method classifies $\mathbb C$-boundaries up to 6 crossings.

We prove a cabling formula for the concordance invariant $ν^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.

In this paper we prove that the closed $4$-ball admits non-Kähler complex structures with strictly pseudoconcave boundary. Moreover, the induced contact structure on the boundary $3$-sphere is overtwisted.

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…

The paper shows links with 2 components are not smoothly slice in a specific 4-manifold.

The paper classifies algebraic curves in 4-balls and their boundaries.

We show that the torus knot $T_{4,9}$ bounds a smooth Möbius band in the $4$-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance inv…

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $γ_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $γ^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $γ^0_X$ is a generalizati…

We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain links in $S^1 \times S^2$, we compute the invariant by reinterpreting it in terms of Hochschild homology. As applications…

Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.

We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces.