A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
The paper defines and calculates Euler characteristics for quandles.
New symmetric quandles constructed from group elements and subgroups.
Survey of recent developments in racks and quandles.
New algebraic structures for topological pairs.
New link colorings using quandle rings and idempotents are stronger than existing methods.
Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial -cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic condit…
We define a quandle variety as an irreducible algebraic variety endowed with an algebraically defined quandle operation . It can also be seen as an analogue of a generalized affine symmetric space or a regular -manifold in algebraic geometry. Assume that is normal as an algebraic variety and that the a…
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 quandle can be described by using a pair of maps called an MCQ Alexander pair, …
In this paper we describe methods for computing rack and quandle cohomology. We illustrate these methods by completely determining the cohomology of prime dihedral quandles.
We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie 2-algebra' to be a 2-vector space equipped with a skew-symmetric bilinear functor satis…
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 which generalize the quandle counting invariant.
A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the c…
Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some p…
New invariant for virtual links using multi-switches and algebraic systems.
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
Embed spherical quandles into Lie groups smoothly.
Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.
Algorithms are described and Maple implementations are provided for finding all quandles of order , as well as computing all homomorphisms between two finite quandles or from a finitely presented quandle (e.g., a knot quandle) to a finite quandle, computing the automorphism group of a finite quandle, etc. Several of…
We define a class of quandle-like structures called pseudoquandles and analyze some of their algebraic properties.
Classifies links with finite N-quandles for some N.
Study metrics on quandles, a knot theory algebraic system.
Study of profinite quandles with constructions and characterizations.
Generalizes quandle constructions and defines a multiplication that results in an abelian group.
Study satellite knots and their quandles related to incompressible tori.
New knot quandle structure for twist-spun trefoils discovered.
We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of quandle homomorphisms from a knot quandle associated to the base to a quandle associated to a fiber. We consider the cases of the monodromy of a branched covering, braid monodro…
The theory of rack and quandle modules is developed - in particular a tensor product is defined, and shown to satisfy an appropriate adjointness condition. Notions of free rack and quandle modules are introduced, and used to define an enveloping object (the `rack algebra' or `wring') for a given rack or quandle. These …
Paper introduces tensor product of quandles for knot classification.
The paper explores new quandle systems for handlebody-links and spatial graphs.
Study algebraic invariants of multi-virtual links.
The paper explores zero-divisors and idempotents in quandle rings, proving their absence in certain cases.
New axioms for singquandles simplify applications and reveal algebraic aspects.
Paper provides criteria to detect non-admissible quandles via coloring.
The paper studies quandles over a hyperboloid and computes a knot invariant.
New quandles classify surfaces, with infinite cohomology.
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 purpose is to give a short introduction to the subject and a guide to the applicat…
We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples.
Paper describes how to extend multiple conjugation quandles using maps.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each …
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 enhancement is stronger than the associated unenhanced counting invariant.
The study enumerates virtual quandles up to isomorphism.
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…
This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…
The paper studies quandles from symmetric group automorphisms and finds a correspondence with conjugacy classes.
The paper studies quandles of hyperbolic 3-space isometries.