arXiv research

New polynomials defined for virtual knots, calculated up to crossing 4.

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersecti…

New polynomial invariants defined for long virtual knots.

In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…

Study intersection polynomials of long virtual knots with supporting genera.

We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called…

Paper introduces skew-symmetric matrices for virtual doodle classification.

Virtual knots with same writhe polynomial have equivalent intersection graphs.

Virtual invariants defined from sheaves on surfaces.

Upper bounds found for Seiberg-Witten moduli spaces under specific conditions.

Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.

A virtual string is a scheme of self-intersections of a closed curve on a surface. We introduce virtual strings and study their geometric properties and homotopy invariants. We also discuss connections between virtual strings, Gauss words, and virtual knots.

Updated polynomial for virtual tangles, compatible with decompositions.

A virtual $n$-string $α$ is a collection of $n$ oriented smooth generic loops on a surface $M$. A stabilization of $α$ is a surgery that results in attaching a handle to $M$ along disks avoiding $α$, and the inverse operation is a destabilization of $α$. We consider virtual $n$-strings up to virtual homotopy, i.e., seq…

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fi…

Develops a method to compute Morse homology for clean but not necessarily transverse intersections.

A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connecti…

The flyping theorem is extended to virtual links and surfaces.

New parities defined on virtual knots linked to crossing indices.

The paper generalizes Segre and Verlinde numbers for surfaces with holomorphic 2-forms.

New invariant for virtual links defined using homology.

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classic…

The paper extends CF-moves to classify virtual links of any number of components.

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard (p,q)-torus knot diagram by replacing all crossings o…

Proves Alexander and Markov theorems for higher genus virtual doodles.

Given a prime $p$, a group is called residually $p$ if the intersection of its $p$-power index normal subgroups is trivial. A group is called virtually residually $p$ if it has a finite index subgroup which is residually $p$. It is well-known that finitely generated linear groups over fields of characteristic zero are …

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number…

Paper proves hardness of learning various complex models under local pseudorandom generators.

Connected sum affects crossing numbers of flat virtual knots.

We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We…

Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.

Study uses sentiment analysis to predict cryptocurrency token returns in virtual reality.

We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot …

Generalizes meander diagrams to virtual knots and introduces new invariants.

We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $ζ$ meaning an analogy with $ζ$-polynomial for virtual links. A degree of $ζ$-polynomial estimates a virtual crossing number. We describe some application of $ζ$-polynomial for the study of m…

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the …

Paper studies pure virtual twin groups and their automorphisms.

Minimal crossing virtual links have minimal supporting genus.

Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram …

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

New invariant connects virtual and classical linking numbers.

We prove that a closed arithmetic hyperbolic 3-manifold with positive first betti number has virtually infinite first betti number.

The paper classifies virtual links using the arc shift operation.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

The paper calculates bridge numbers for knots using machine learning.

Novel approach for large genus intersection number asymptotics.