Related papers: Generalised Biracks and the Birack Polynomial Inva…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used…
We present an elementary introduction to one of the most important today knot theory approaches, which gives rise to a representation for a class of knot polynomials in terms of quantum groups. Historically, the approach was at the same…
We introduce an infinite family of quantum enhancements of the biquandle counting invariant we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a…
Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and…
In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these…
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…
We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of…
In this paper, we consider biquandle colorings for knotoids in $\mathbb{R}^2$ or $S^2$ and we construct several coloring invariants for knotoids derived as enhancements of the biquandle counting invariant. We first enhance the biquandle…
The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo…
Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the…
An open bosonic string is considered with the aim to construct a general gauge invariant, being a polynomial of Fubini-Veneziano (FV) fields. The FV fields are transformed as 1-forms on $S^1$, that allows to formulate the problem in…
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character…
This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat…
We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case…
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones…
Let $\mathsf{B}_1$ be the polynomial ring $\mathbb{C}[a^{\pm1},b]$ with the structure of a complex Hopf algebra induced from its interpretation as the algebra of regular functions on the affine linear algebraic group of complex invertible…