Related papers: Multivariate Alexander quandles, VI. Metabelian gr…
Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is…
The aim of this paper is to provide a new characterization of isomorphism classes of generalized Alexander quandles in terms of the underlying groups and their automorphisms. This extends the previous result [4, Theorem 1.4]. Additionally,…
Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare this $K_1$-class with other Alexander polynomials. In terms of…
In this paper we conjecture that the Links-Gould invariant of links, that we know is a generalization of the Alexander-Conway polynomial, shares some of its classical features. In particular it seems to give a lower bound for the genus of…
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…
We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we…
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…
In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free…
Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted…
We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The…
Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The…
We completely describe good involutions of free quandles and subquandles of twisted conjugation quandles of groups, including all Alexander quandles. As an application, we enumerate good involutions of linear quandles, and we provide…
In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among…
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…
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…
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…
We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We…
A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation…
We are intereseted in quandles and their enveloping groups. Various results are proven. We show that a quandle $Q$ and its image in the enveloping group $G(Q)$ have isomorphic enveloping groups. The image quandle is injective. For $Q$ a…
We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be…