相关论文: Virtual Knot Theory

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of…

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof…

We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both…

In this paper we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain…

For a knot diagram $K$, the classical knot group $\pi_1(K)$ is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this…

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there…

In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus $g_{vc}(K)$ and the virtual bridge number $vb(K)$ invariants of virtual knots. One can see from the definitions that for an classical knot $K$…

We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and…

We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce…

The aim of this survey article is to highlight several notoriously intractable problems about knots and links, as well as to provide a brief discussion of what is known about them.

A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By…

We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental…

Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Chern-Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots…

The fundamental problem of knot theory is to know whether two knots are equivalent or not. As a tool to prove that two knots are different, mathematicians have developed various invariants. Knots invariants are just functions that can be…

A book Chapter consisting of some of the main areas of research in graph theory applied to physics. It includes graphs in condensed matter theory, such as the tight-binding and the Hubbard model. It follows the study of graph theory and…

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a…

We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 48 inequalities between 33 knot…

A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…

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.

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of…