arXiv research

Every knot can be transformed into an unknot without cutting.

We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at least 2^{2n-1} non-isotopic minimal spanning surfaces of genus n.

New gauge theory invariant detects non-smooth isotopy of $P^2$-knots.

We prove that two Legendrian knots in a contact structure which is trivializable as a plane bundle are Legendrian isotopic provided that (1) they are isotopic as framed knots, (2) they have the same rotation number with respect to some parallelization of the contact structure, and (3) there is an overtwisted disk disjo…

We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.

New knots bound multiple non-isotopic ribbon disks.

The study limits the number of 2-holed tori in knot exteriors.

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifo…

Manufacturing infinite sets of knotted and unknotted surfaces in 4-manifolds.

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergenc…

Knots in 3-manifolds are equivalent if isotopic, except in special cases.

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

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.

The $n$-skeleton of the canonical cubulation $\cal C$ of $\mathbb{R}^{n+2}$ into unit cubes is called the {\it canonical scaffolding} ${\cal{S}}$. In this paper, we prove that any smooth, compact, closed, $n$-dimensional submanifold of $\mathbb{R}^{n+2}$ with trivial normal bundle can be continuously isotoped by an amb…

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive hom…

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

The abstract discusses knots and their isotopy to the unknot, proving a specific case.

New examples show transverse knots are determined by their branched covers.

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…

New findings show that some knotted surfaces remain distinct even after many stabilizations.

It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum volume of any link $L$ in a natural class of alternating, hyperbolic links such tha…

We prove that any knot of $\mathbb{R}^3$ is isotopic to a Fourier knot of type $(1,1,2)$ obtained by deformation of a Lissajous knot.

Study shows hyperbolic knots' monodromy without fixed points.

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then either \begin{itemize} \item $d(P)\leq 2-χ(Q-K)$, or \item K can be isotoped to be disjoint from Q so that after the isotopy Q is a He…

Study finds non-isotopic transverse tori in Engel manifolds.

Links can be transformed into many others using a specific operation.

We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.

We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curva…

New examples of non-simple knots in Lens spaces show rich botany.

We exhibit an encoding of knots into processes in the π-calculus such that knots are ambient isotopic if and only their encodings are weakly bisimilar.

Introduces a Cost function to measure Legendrian knot obstructions.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some $Ω_3$-moves that only differ in how the three strands that are involved in the move are ordered on the knot. To transform knot diagrams of isotopic knots into each other …

New method to classify simple Smale flows on $S^3$.

This paper classifies 2-plat 2-knots using a new invariant.

New proof of link factorization theorem, avoiding case exhaustion.

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot $K$ is $n$-adjacent to a fibered knot $K'$, for some $n>1$, then either the genus of $K$ is larger than that of $K'$ or $K$ is isotopic to $K'$.

Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian …

The paper links knot properties in projective space to their fundamental groups.

A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to answer th…

We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing Legendrian knots that have the same classical invariants but are not Legendrian isotopic.

New examples of knotting phenomena in 4-manifolds with specific fundamental groups.

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…

A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are isotopic as transverse knots are also isotopic as transverse braids. The methods of the…

In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.

An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…

Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…