Related papers: Link invariants from finite biracks
We introduce and investigate dichromatic singular links. We also construct G-Family of singquandles and use them to define counting invariants for unoriented dichromatic singular links. We provide some examples to show that these invariants…
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of…
We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have…
In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As…
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…
Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The…
Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology…
This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.
In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schr\"{o}er--the \emph{bridge quiver} whose vertices are (representatives of cyclic…
We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.
We introduce the notion of a hierarchical quandle, which is a generalisation of diquandles and multi-quandles. Using hierarchical quandle colourings, we construct a cocycle invariants for links coloured by quandles.
We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a…
We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of…
We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding…
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained…
In this short survey we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent…
Given a biquandle $(X, S)$, a function $\tau$ with certain compatibility and a pair of {\em non commutative cocyles} $f,h:X \times X\to G$ with values in a non necessarily commutative group $G$, we give an invariant for singular knots /…
For a compact smooth manifold with corners (or finite CW-complex) $X$, we can prescribe a finite set of spin or spin$^h$ manifolds (possibly with boundary) mapping into it so that every real vector bundle over $X$ is determined, up to…
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding…
I propose a notation for biracks that includes from the begining the knowledege of the associated (or underlying, or derived) rack structure. Motivated by results of Rump in the involutive case, this notation allows to generalize some…