Related papers: Biquandles, quivers and virtual bridge indices

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which…

Introduced in arXiv:2211.12606, biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian…

The forbidden moves in virtual knot theory can be used to unknot any knot, virtual or classical; however, multi-component crossings in links can still survive, resulting a fused link. Using the forbidden moves, we categorify fused links…

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation…

Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…

In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind…

Flat virtual links are some variant of links, and semiquandles are counterparts of quandles or biquandles, which axiomize the Reidemeister-like moves. In this paper, we give some example of semiquandle and introduce an invariant for flat…

Virtual quandles with two operations are discussed in the article. Certain knot invariant is constructed and used to distinguish two long virtual knots.

This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram…

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also…

The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links…

In this paper, we introduce biquandle power brackets, an infinite family of invariants of oriented links containing the classical skein invariants and the quandle and biquandle 2-cocycle invariants as special cases. Biquandle power brackets…

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut…

We consider the question of which virtual knots have finite fundamental medial bikei. We describe and implement an algorithm for completing a presentation matrix of a medial bikei to an operation table, determining both the cardinality and…

We use the structure of skew braces to enhance the biquandle counting invariant for virtual knots and links for finite biquandles defined from skew braces. We introduce two new invariants: a single-variable polynomial using skew brace…

We extend the Wirtinger number of links, an invariant originally defined by Blair, Kjuchukova, Velazquez, and Villanueva in terms of extending initial colorings of some strands of a diagram to the entire diagram, to spatial graphs. We prove…

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

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…

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…

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives…