相关论文: Stable Equivalence of Knots on Surfaces and Virtua…

The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are…

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably…

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that…

This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…

In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define…

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this…

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the…

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the…

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter…

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures called birack dynamical cocycles. The new…

It is known that the Kauffman-Murasugi-Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with…

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic…

Virtual knot theory has experienced a lot of nice features that did not appear in classical knot theory, e.g., parity and picture-valued invariants. In the present paper we use virtual knot theory effects to construct new representations of…

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that…

(a) We propose a ``static'' construction procedure for random networks with given correlations of the degrees of the nearest-neighbor vertices. This is an equilibrium graph, maximally random under the constraint that its degree-degree…

In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant…

The relationship between stable holomorphic vector bundles on a compact complex surface and the same such objects on a blowup of the surface is investigated, where "stability" is with respect to a Gauduchon metric on the surface and…

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the…