Related papers: Cubulated moves for 2-knots

Let $X$ be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on $X$ are isotopic. This implies that…

We introduce a local deformation called the virtualized $\Delta$-move for virtual knots and links. We prove that the virtualized $\Delta$-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient…

We say that a topological $n$-manifold $N$ is a cubical $n$-manifold if it is contained in the $n$-skeleton of the canonical cubulation $\mathcal{C}$ of ${\mathbb{R}}^{n+k}$ ($k\geq1$). In this paper, we prove that any closed, oriented…

We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex.…

We consider a diagrammatic approach to investigate tame knots and links in three dimensional torus $T^3$. We obtain a finite set of generalised Reidemeister moves for equivalent links up to ambient isotopy. We give a presentation for…

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through…

We study the gordian graph of all knots in $\R^3$: two knots are adjacent if they differ by a single crossing change. We prove that this graph contains isometrically an infinite countable tree with infinite valency, and that the complement…

When does the double cover of the three-sphere branched along an alternating link bound a rational homology ball? Heegaard Floer homology generates a necessary condition for it to bound: the link's chessboard lattice must be cubiquitous,…

We describe a procedure to construct infinite sets of pairwise smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some…

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson-Gordon obstructions. Similarly all…

Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister…

In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only…

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal…

2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known…

Recently Swatee Naik and Theodore Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to…

We consider commuting pairs of holomorphic endomorphisms of P^2 with disjoint sequence of iterates. The remaining case to be studied is when their degrees coincide after some number of iterations. We show in this case that they are either…

By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence…

For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the…

We prove that two closed oriented 3-manifolds have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function) if, and only if, they belong to the same class of a certain surgery…

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz…