arXiv research

Standard trisection diagrams found for Mazur type 4-manifolds.

Study obstructs symplectic structures on Mazur manifolds.

Construct non-split 2-component links in $S^4$ to produce exotic pairs of 4-manifolds

In this paper we find infinitely many Mazur type manifolds and corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also find such manifolds among 4-manifolds constructed from Bing's house. Our manifolds …

Mazur's knot exterior is described by a single regular ideal octahedron, leading to hyperbolic structures related to the Whitehead link.

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…

We prove that for every $n>2$, the Banach-Mazur compactum Q(n) is the compactification of a Hilbert cube manifold by the Euclidean point. For $n=2$ this result was proved earlier.

We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.

In this note, we prove that if the boundary of a Mazur-type $4$-manifold is an irreducible Heegaard Floer homology $L$-space, then the manifold must be the $4$-ball, and the boundary must be the $3$-sphere. We use this to give a new proof of Gabai's Property R.

New method uses Khovanov homology to distinguish exotic 4-manifolds.

The paper constructs contractible manifolds with knotted spheres.

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

New examples of Kirby-Ramanujam spheres found, leading to contractible 4-manifolds.

Study of knots with generalized Mazur patterns and their invariants.

Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.

Study Mazur doubles of knots and their relation to the Slope Conjecture.

The Mazur pattern acts by the identity up to topological concordance.

New infinite-rank summand found in knot concordance group.

We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed $3$-manifolds. We first prove that, given $Y^3 \neq S^3$, for any non-trivial element $g\in π_1(Y)$ there are infinitely many distinct smooth almost-concordance classes in the free homoto…

New corks found that are not strong and exotic.

We extend to the framework of locally $L^0$-convex modules some results from classical convex analysis. Namely, randomized versions of Mazur lemma and Krein-Smulian theorem under mild stability properties are provided.

New link detection results using closures of 3-braids.

We show that a certain involution on the well known homology sphere (the boundary of the Mazur manifold) induces a nontrivial homomorphism on its Heegard-Floer homology groups (recently defined by Ozsvath and Szabo). We discuss a possible application of this to constructing exotic smooth structures on 4-manifolds.

Homotopy proof for pseudomanifolds via branched covers.

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…

Lickorish has constructed large families of contractible 4--manifolds that have knotted embeddings in the 4--sphere and has also shown that every finitely presented perfect group with balanced presentation occurs as the fundamental group of the complement of a knotted contractible manifold. Here we make a few observati…

We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an infinite family $\{(W_{n,k},W'_{n,k})\}_{k=1}^{\infty}$ of exotic pairs of $4$-manifold…

New knots not slice in rational 4-balls found.

Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic…

Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.

We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsvath--Szabo invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard--Floer homologies. We also show …

In a preceding work it is determined when a centrally symmetric convex body in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ Consequently, the class of tensorial bodies is introduced, an associated tensorial Banach-Mazur d…

We formalize the arithmetic topology, i.e. a relationship between knots and primes. Namely, using the notion of a cluster C*-algebra we construct a functor from the category of 3-dimensional manifolds M to a category of algebraic number fields K, such that the prime ideals (ideals, resp.) in the ring of integers of K c…

Stability result for nearly isometric subspaces and Finsler surfaces.

Symplectic capacities of domains near balls are well-defined, but not for all $C^1$-close domains.

We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and Mather on homeomorphism groups.

Develops potential theory for WZW equation in Kähler potentials space.

Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of …

Given two compact n-dimensional manifolds in the smooth, piecewise linear or topological categories, basic results of B. Mazur and others give simple criteria for determining whether their products with Euclidean spaces of sufficiently large dimension are isomorphic in the given category. This paper studies such questi…

We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong…

New 4-manifold examples show necessary conditions for 4D Light Bulb Theorem.

We discuss the relationship between two analogues in a 3-manifold of the set of prime ideals in a number field. We prove that if $(K_i)_{i\in \mathbb{N}_{>0}}$ is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then $\mathcal{K}=\cup_i K_i$ is a stably generic link in the sense of Mih…

Study on the topology of tensorial bodies, showing they are homeomorphic to a product space.

New representation of PSL2(R) on infinite hyperbolic space via convex bodies.

A \emph{generalized dunce hat} is a 2-dimensional polyhedron created by attaching the boundary of a disk $Δ$ to a circle $J$ via a map $f:\partial Δ\to J$ with the property that there is a point $v \in J$ such that $f^{-1}(\{v\})$ is a finite set containing at least 3 points and $f$ maps each component of $\partial Δ- …

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…

This paper proves an upper limit on rational points on curves.