Related papers: Biquandle Bracket Quivers
We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However,…
In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic…
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type…
In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and…
We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$…
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of…
The number of colorings of a knot diagram by a quandle has been shown to be a knot invariant by CJKLS using quandle cohomology methods. In a previous paper by the second named author, the CJKLS invariant was refined and, in particular, it…
While knotoids on the sphere are well-understood by a variety of invariants, knotoids on the plane have proven more subtle to classify due to their multitude over knotoids on the sphere and a lack of invariants that detect a diagram's…
The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be…
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a…
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram.…
We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of…
We introduce a modified homology and cohomology theory for involutory biquandles (also known as \textit{bikei}). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and…
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 group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.
We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as…
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular…
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…
We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose…