Related papers: G-Family Polynomials
We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family…
We introduce the notion of a $G$-family of quandles which is an algebraic system whose axioms are motivated by handlebody-knot theory, and use it to construct invariants for handlebody-knots. Our invariant can detect the chiralities of some…
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 define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants…
We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in…
A $G-$family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate…
We introduce two new families of polynomial invariants of oriented classical and virtual knots and links defined as decategorfications of the quandle coloring quiver. We provide examples to illustrate the computation of the invariants, show…
We introduce a generalization of the quandle polynomial. We prove that our polynomial is an invariant of stuquandles. Furthermore, we use the invariant of stuquandles to define a polynomial invariant of stuck links. As a byproduct, we…
We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the…
Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite…
We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction…
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…
We introduce a six-variable polynomial invariant of Niebrzydowski tribrackets analogous to quandle,rack and biquandle polynomials. Using the subtribrackets of a tribracket, we additionally define subtribracket polynomials and establish a…
We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and…
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 a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants…
An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for…
We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not classified by their generalized quandle…
Quandle Coloring Quivers are directed graph-valued invariants of classical and virtual knots and links associated to finite quandles. Quandle action quivers are subquivers of the full quandle coloring quiver associated to quandle actions by…
In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.