Related papers: Link invariants from finite biracks
We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as…
Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the…
This article surveys many aspects of the theory of quandles which algebraically encode the Reidemeister moves. In addition to knot theory, quandles have found applications in other areas which are only mentioned in passing here. The main…
In the preprint of V. Bardakov, T. Kozlovskaya, D. Talalaev (Self-distributive bialgebras, arXiv:2501.19152) it was formulated a problem of classification of self-distributive bialgebras and was given classification of two-dimensional…
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…
This paper has partially a novel and partially a survey character. We start with a short review of rack (two term) homology of self distributive algebraic structures (shelves) and their connections to knot theory. We concentrate on a…
We investigate connections between biquandle colorings, quiver enhancements, and several notions of the bridge numbers $b_i(K)$ for virtual links, where $i=1,2$. We show that for any positive integers $m \leq n$, there exists a virtual link…
Generalising a recent work of Dequ\^ene et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is…
We define enhancements of the quandle counting invariant for knots and links with a finite labeling quandle Q embedded in the quandle of units of a Lie algebra \mathfrak{a} using Lie ideals. We provide examples demonstrating that the…
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 denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers…
The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group \lambda(M) measures certain…
The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the…
In these notes we collect some results about finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ and related invariants of framed tangles which are well-known to experts but difficult to find in the literature. In particular, we…
The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove…
We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other…
We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.
We define a sequence of integer-valued invariants $\gamma^k(L)$ for a $3$-component link $L$. We prove that the resulting $\gamma$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift…
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…
We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide…