Related papers: Quandle coloring quivers and 2-bridge links
Biquandles are generalizations of quandles. As well as quandles, biquandles give us many invariants for oriented classical/virtual/surface links. Some invariants derived from biquandles are known to be stronger than those from quandles for…
To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the {\em involutory} quandle). Hoste and Shanahan \cite{HS2}…
We introduce the notion of mc-biquandles, algebraic structures which have possibly distinct biquandle operations at single-component and multi-component crossings. These structures provide computable homset invariants for classical and…
We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle…
In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle…
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting invariant in the sense of defining…
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 their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket…
Given a quandle, we can construct a symmetric quandle called the symmetric double of the quandle. We show that the (co)homology groups of a given quandle are isomorphic to those of its symmetric double. Moreover, quandle coloring numbers…
A biquandle is a solution to the set-theoretical Yang-Baxter equation, which yields invariants for virtual knots such as the coloring number and the state-sum invariant. A virtual biquandle enriches the structure of a biquandle by…
Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this…
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror…
In this paper, we extend the notion of directed clique complex to quivers and introduce an associated homology theory. By applying this construction to biquandle coloring quivers, we obtain new invariants of links. We then introduce a…
We define a family of quiver representation-valued invariants of oriented classical and virtual knots and links associated to a choice of finite quandle $X$, abelian group $A$, set of quandle 2-cocycles $C\subset H^2_Q(x;A)$, choice of…
We show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the $\Hom$…
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p,…
The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by…
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…
In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones…
In the first part of this paper, we present general results concerning the colorability of torus knots using conjugation quandles over any abstract group. Subsequently, we offer a numerical characterization for the colorability of torus…