arXiv research

New method computes knot Floer homology for satellite knots.

We study the 3-dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial $(2p,q)$-torus or $(2p,q)$-cable knot. We show that unlik…

Study minimum ribbonlength of immersed flat knots and links.

The paper tackles deeply slice knots via immersed curves.

A new method converts knot Floer homology to immersed curves.

New knots not slice in rational 4-balls found.

Satellite knots with (1,1)-patterns have their Floer homology computed using immersed curves.

The study classifies immersed surfaces with knot group Z in simply-connected 4-manifolds.

A new mosaic system for immersed surface-links is introduced.

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…

Lower bounds on unknotting number for cabled knots.

New invariant shows fifth move is unique and extends biquandle theory for surface-links.

Knot Floer homology is reinterpreted as immersed curves.

We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if $\frac 1{q_1-1} + \frac 1{q_2-1} + \frac 1{q_3-1} \leq 1$ then a Montesinos knot $K(\frac{p_1}{q_1}, \frac{p_2}{q_2}, \frac{p_3}{q_3})$ admits no atoroidal Seifert fibered surgery.

New proof shows most thin knots satisfy Cabling Conjecture.

Study corrects previous work on knot Floer homology of certain pretzel knots.

New insights into knot fusion numbers via cabling.

Quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

All knots with unknotting number ≤ 21 are smoothly slice in K3 surface.

This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an embedding or immersion. We then discuss recent work that creates invariants of immersio…

We study three knot invariants related to smoothly immersed disks in the four-ball. These are the four-ball crossing number, which is the minimal number of normal double points of such a disk bounded by a given knot; the slicing number, which is the minimal number of crossing changes to a slice knot; and the concordanc…

We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10% of the knots in Rolfsen's table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots…

The study examines how twisting a knot affects its homology and stability properties.

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…

Smooth curves with specific curvature can be closely approximated.

Study shows spherical embedding and immersion components are related to homotopy groups.

In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formu…

We prove that every immersed $C^2$-curve $γ$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_γ$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_γ$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary p…

A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.

We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a $3$-manifold $M$ that are transverse to a nowhere-zero vector field $V$ up to the corresponding isotopy relation. Such knots are called …

Study proves h-principles for curves in bracket-generating distributions.

The paper details folding of branched covers of the 3-sphere over knots.

Concordance invariants of knots are derived from the instanton homology groups with local coefficients, as introduced in earlier work of the authors. These concordance invariants include a 1-parameter family of homomorphisms $f_{r}$, from the knot concordance group to the reals. Prima facie, these concordance invariant…

32 knot projections classified based on forbidden Reidemeister moves.

Classifies certain 3D knots with specific properties.

We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the asso…

We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, conside…

Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$ character varieties of 2-tangle decompositions of knots. Calculations for torus knots…

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study…

Any knot in $S^3$ may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that k…

Proves Alexander and Markov theorems for higher genus virtual doodles.

A singular knot is an immersed circle in $\mathbb R^{3}$ with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing filtration on the infinite dimensional vector space spanned by isotopy classes …

In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology.…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map fr…

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants of wave fronts on a surface $F$ is isomorphic to the group of Vassiliev invariant…

Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the projection of an n-knot which is diffeomorphic to the standard sphere. We prove the…

Curves with constant torsion can be deformed arbitrarily.