arXiv research

This paper studies the homotopy type of the moduli space of compact n-manifold thickenings of a finite complex. The main result computes the homotopy fibers of the stabilization map from n-thickenings to (n+1)-thickenings in a wide range. In particular, we obtain an EHP-type sequence for thickenings extending work of C…

New homotopy types defined for links in thickened surfaces with higher genus.

Geometrically interprets virtual knotoids in thickened surfaces.

Defines homotopy type for links in thickened surfaces.

Homotopy types of Vietoris-Rips metric thickenings of the circle confirmed.

Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.

The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.

Hyperbolic links in thickened torus decompose into angled tetrahedra.

Meridian lemma extended to fully alternating links in thickened surfaces.

New Khovanov homology for links with multiple punctures.

Study fully augmented links in thickened torus, generalizing $S^3$ results.

A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow Σ\times [0,1]$, where $Σ$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classic…

This paper generalizes octahedral decomposition to links in thickened surfaces.

Homotopy equivalence shown between complex and thickened versions of manifolds.

Spaces over BO are equivalent to thickened manifolds.

A chord index homomorphism for knots in thickened surfaces is constructed.

The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…

Establishes bounds on Andrews-Curtis moves for trivial group presentations.

Analog of Kauffman bracket for non-orientable knots in thickened surface.

A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.

Extends Heisenberg homology to ribbon graphs.

The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.

We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995…

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable…

Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometr…

A cobordism between links in thickened surfaces consists of a surface $S$ and a $3$-manifold $M$, with $S$ properly embedded in $M \times I$. We show that there exist links in thickened surfaces such that if $(S,M)$ is a cobordism between them in which $S$ is simple, then $M$ must be complex. That is, there…

The paper extends knotoid theory to annular and toroidal settings.

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more general…

An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual …

Decomposes string links in a surface into prime components.

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…

Study links' flat-virtual diagrams to create link invariants.

Study invariants of null-homologous knots in thickened surfaces.

Algorithm converts plat to standard closure of braids in 3D and related spaces.

We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened $3$-manifolds. In particul…

The study determines conditions for hyperbolicity of links in thickened surfaces with boundary.

The paper develops a method to map knots in a cylinder to virtual-flat knots.

Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thu…

New invariants for virtual knots via spanning surfaces.

This paper classifies periodic weaves and their universal cover, extending Tait's conjectures.

Almost forty years ago, C.T.C. Wall systematically analyzed the set of "thickenings" of a finite CW complex. Of the results he obtained, probably the most computationally important is the "suspension theorem," which is an exact sequence relating the n-dimensional thickenings of a finite complex to its (n+1)-dimensional…

The paper explores group presentations for links in thickened surfaces, proving their relationship and introducing new invariants.

Study on knots in contact manifolds, focusing on their width and thickness.

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

Totally geodesic surfaces found in knots and links.

New algebra invariant distinguishes Legendrian knots in convex surfaces.

This paper studies knots in a thickened surface and introduces a new way to label crossings.

We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…