In this paper, we prove Gukov-Pei-Putrov-Vafa's conjecture that the Witten-Reshetikhin-Turaev invariants are radial limits of homological blocks, which are $ q $-series introduced by them for...

Every smooth homotopy 4-sphere is diffeomorphic to the 4-sphere.

In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to...

The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them such that the two knot meridians are in the...

We give an infinite family of embeddings of $\mathbb{R} P^2$ to $S^4$ such that they are mutually topologically isotopic however are not smoothly isotopic to each other. Moreover, they are...

We define an involutive version of Khovanov homology and a pair of integer-valued invariants $(\underline{s}, \bar{s})$ for strongly invertible knots, which is an equivariant version of the...

Our aim is to determine the regular homotopy classes of immersions related to Arnol'd's simple singularities. For every type of simple singularities, we determine the regular homotopy class of the...

By using grid homology theory, we give an explicit algorithm for computing Ozsváth-Stipsicz-Szabó's $Υ$-invariant and the $d$-invariant of Dehn surgeries along knots in $S^3$. As its...

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they...

A Divide with cusps is the image of a proper generic immersion from finite intervals and circles into a $2$-disk which allows to have cusps. A divide with cusps is the generalization of the notion...

One strategy for distinguishing smooth structures on closed $4$-manifolds is to produce a knot $K$ in $S^3$ that is slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic...

We study the Milnor fiber boundary for hyperplane arrangements in $\mathbb{C}^3$. This is one of the examples of non-isolated surface singularities, which are studied by Némethi--Szilárd. In...

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves...

A musical instrument, the plastic hormonica, is defined here as a birthday present for Dennis Sullivan, who pioneered and helped popularize the hyperbolic geometry underlying its construction....

Given an irreducible, end-periodic homeomorphism f of a surface S with finitely many ends, all accumulated by genus, the mapping torus is the interior of a compact, irreducible, atoroidal...

We give a family of slice-torus invariants $\tilde{ss}_c$, each defined from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime...

We develop a version of Seiberg--Witten Floer cohomology/homotopy type for a spin$^c$ 4-manifold with boundary and with an involution which reverses the spin$^c$ structure, as well as a version of...

It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth 4-dimensional manifolds are trivial.

The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev's shadows, which, roughly speaking, measures how complicated a $2$-skeleton...

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold...

We present a framework for studying transverse knots and symplectic surfaces utilizing the Seiberg-Witten monopole equation. Our primary approach involves investigating an equivariant...

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering...

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. Moreover, we prove a gluing formula relating our invariant with the first author's...

In this paper, we prove a conjecture by Gukov-Pei-Putrov-Vafa for a wide class of plumbed 3-manifolds. Their conjecture states that Witten-Reshetikhin-Turaev (WRT) invariants are radial limits of...

Given a hyperbolic surface $Σ$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for...

A surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a...

This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and...

Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and...

Here we give a concrete description of the cork automorphism $f:\partial W\to \partial W$ of the infinite order loose-cork $(W,f)$, defined in \cite{a2}. It is obtained by concatenating the...

We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth...

Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings. Two graph homologies have been invented from two approaches to study the space of long...

This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.

We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i.e. they can be given handle decompositions...

We show that the knot group of any knot in any integer homology sphere admits a non-abelian representation into $SU(3)$ such that meridians are mapped to matrices whose eigenvalues are the three...

The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems...

In this paper, we consider the non-singular extension problem of circle-valued Morse functions on closed orientable surfaces. The problem is when can a submersion from a collaring of a closed...

The Upsilon invariant of a knot is a concordance invariant derived from knot Floer homology theory. It is a piecewise linear continuous function defined on the interval $[0,2]$. Borodzik and...

For an unpunctured marked surface $Σ$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},Σ}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $Σ$ with the boundary skein...

For a negative definite plumbed three-manifold, we give an integral representation of the appropriate average of the GPPV invariants of Gukov--Pei--Putrov--Vafa, which implies that this average...

Let $g$ be a non-negative integer, $Σ_g$ a closed orientable surface of genus $g$, and $\mathcal{M}_g$ its mapping class group. We classify all the group homomorphisms $π_1(Σ_g)\to...

In this note we make progress toward a conjecture of Durham--Fanoni--Vlamis, showing that every infinite-type surface with finite-invariance index 1 and no nondisplaceable compact subsurfaces...

We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored...

Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a 4-dimensional analogue to bridge decompositions of classical links, which gives a numerical invariant of surface links...

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...

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{Σ,\mathbb{W}}^{A}$ of a walled surface...

Since the Teichmüller space of a surface $R$ is a deformation space of complex structures defined on $R$, its Bers boundary describes the degeneration of complex structures in a certain sense....

The knot quandle is a complete invariant for oriented classical knots in the $3$-sphere up to orientation. Eisermann computed the second quandle homology group of the knot quandle and showed that...

Given a symplectic 4-manifold $(X,ω)$ with rational symplectic form, Auroux constructed branched coverings to $(CP^2,ω_{FS})$. By modifying a previous construction of...

A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear...

We show the existence of a $4$-manifold with boundary that admits two non-diffeomorphic minimal genus relative trisections of the same $(g,k;p,b)$-type. To prove this, we introduce a simple...

These are the lecture notes for a course on Heegaard Floer homology held at PCMI in Summer 2019. We describe Heegaard diagrams, Heegaard Floer homology, knot Floer homology, and the relationship...

Given a link of a normal surface singularity with its canonical contact structure, we compare the collection of its Stein fillings to its Milnor fillings (that is, Milnor fibers of possible...

In previous work, the second author defined 'equivariant instanton homology groups' $I^\bullet(Y,π;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $π$, and a PID $R$. These...

In this paper we study smooth structures on closed oriented 4-manifolds with fundamental group Z_2 and definite intersection form. We construct infinitely many irreducible, smooth, oriented,...

Watanabe disproved the 4-dimensional Smale conjecture by constructing topologically trivial $D^{4}$-bundles over spheres and showing that they are smoothly nontrivial using configuration space...

A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving...

A marked strongly invertible knot is a triple $(K,h,δ)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $δ\subset \operatorname{Fix}(h)\cong S^1$ bounded by...

We describe low dimensional homology groups of $\mathrm{Diff}^δ_+S^1$ in terms of Haefliger's classifying space $B\overlineΓ_1$ by applying a theorem of Thurston. Then we consider the...

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative...

We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two...

In this paper, we introduce a method, called a plat form, of describing a surface-link in the 4-space using a braided surface. We prove that every surface-link, which is not necessarily...

A spatial surface is a compact surface embedded in the $3$-sphere. We assume that each connected component has non-empty boundary. Spatial surfaces are represented by diagrams of spatial trivalent...

A $k$-twist spun knot is an $n+1$-dimensional knot in the $n+3$-dimensional sphere which is obtained from an $n$-dimensional knot in the $n+2$-dimensional sphere by applying an operation called a...

We construct an example of a cork that remains exotic after taking a connected sum with $S^2 \times S^2$. Combined with a work of Akbulut-Ruberman, this implies the existence of an exotic pair of...

Khovanov--Rozansky's HOMFLY homology is determined for all prime knots with up to 11 crossings by direct computations.

The purpose of this paper is to construct non-trivial cocycles of the space $Emb(\mathbb{R}^j, \mathbb{R}^{n})$ of long embeddings. We construct the cocycles by integral over configuration spaces,...

We prove that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology, unlike what happens in all dimensions $\neq 4$. We detect...

Agol proved that ribbon concordance forms a partial ordering on the set of knots in the $3$-sphere. In this paper, we prove that all tight fibered knots are minimal in this partially ordered set....

This is a (very subjective) survey paper for nonspecialists covering group actions on Gromov hyperbolic spaces. The first section is about hyperbolic groups themselves, while the rest of the paper...

We give a description of finitely generated prosoluble subgroups of the profinite completion of $3$-manifold groups and virtually compact special groups.

In this paper, we introduce a notion of a fat CW complex to show that a closed manifold is a regular CW complex, while it is not always the case if we discuss about a smooth CW structure,...

In this survey, we present most recent highlights from the study of the homology cobordism group, with a particular emphasis on its long-standing and rich history in the context of smooth...

Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called...

This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with boundary that are properly embedded in the 4-ball. Our main results provide two different...

This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are...

The classical Poincar{é} conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston's program on geometrizations of...

We describe the spaces of the positive and tropical points of the moduli space $\mathcal{P}_{PGL_2,Σ}$ introduced by Goncharov--Shen [GS19] as certain Teichmüller and lamination spaces,...

We consider the question of extending a smooth homotopy coherent finite cyclic group action on the boundary of a smooth 4-manifold to its interior. As a result, we prove that Dehn twists along any...

We give a first example of 2-knots with the same knot group but different knot quandles by analyzing the knot quandles of twist spins. As a byproduct of the analysis, we also give a classification...

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, right-angled Artin groups, free groups,...

Surgery on a knot in $S^3$ is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of...