arXiv research

Chern-Simons theory on Seifert 3-manifolds evaluated.

Study of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.

New invariants for 3-manifolds from foliations and noncommutative geometry.

Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.

Formula calculates invariant for 3-manifolds with torus boundaries.

3-manifolds' volumes match stable integral values.

Following the analogies between 3-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set $\mathcal{K}$ of knots in a 3-manifold $M$, we first present a local theory for each knot in $\mathcal{K}$, which is analogous to local class field theory, and then,…

Study finds infinite hyperbolic 3-manifolds without fillable contact structures.

The paper develops a new theory for knots and 3-manifolds with involutions.

Alternative proof of flatness for Ricci-pinched 3-manifolds.

Study evaluates Chern-Simons theory on Seifert fibred 3-manifolds.

New invariant unifies two theories of 3-manifolds, recovering quantum invariants.

Survey on 3-manifold problems and their complexity.

Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.

Computes Vafa-Witten invariants of 3-manifolds.

Study of $p$-subgroups in 3-manifold groups' completions.

The volume conjecture is extended to all orders for hyperbolic 3-manifolds using complex Chern-Simons theory.

Culler-Shalen theory extended to arbitrary characteristic.

Categorifies Stokes coefficients in Chern-Simons theory models.

Proof of orientable 3-manifolds parallelizability using knot theory.

Floer theory uses categories to construct 3-manifold invariants.

We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibratio…

Decomposes 3-manifold pairings using twisted Blanchfield pairings.

Geometric methods study 3-manifold splittings.

Homological invariant defined using gauge theory for 3-manifolds.

These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic 3-manifolds and sketch some interesting results in the theory of 3-manifolds (suc…

Develops skein theory for 3-manifolds with defects, extending quantum character stacks.

Novel M-theory approach classifies topological phases of matter.

Power series invariant of hyperbolic 3-manifolds matches knot invariants.

Unified framework for 3D and 4D manifold and knot theory.

Study $Θ$-invariants for spherical 3-manifolds via $\mathbb{Z}π$-homology equivalences.

Algorithm finds essential surfaces in 3D shapes.

This is a problem list in the theory of foliations and laminations of 3-manifolds. The focus is on the relationship of foliations and laminations with other aspects of 3-manifold topology, especially with the Thurston theory of geometric structures. This problem list is not meant to be comprehensive; rather it focusses…

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.

Overview of 3D TQFTs and 3-manifold invariants.

New findings on 3-manifolds using Heegaard Floer theory.

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independ…

New theory couples Chern-Simons to matter, topological.

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain…

We study a topological analogue of idèlic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal{K})$, we intr…

Graphs describe contact surgery on 3-manifolds.

Solves long-standing classification problem for certain 3-manifolds.

The paper classifies tight contact structures on specific plumbed 3-manifolds.

Expository notes on intersection theory of holomorphic curves in 3-manifolds.

Proposes categorification of Z-invariants for specific 3-manifolds.

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.

Uncountably many contractible 3-manifolds with double 3-space property and those without are identified.

A criterion for Whitney disks connects intersections in 3-manifold homology.