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.
The paper connects GL-racks to knot coloring invariants.
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…
The paper introduces a new coloring invariant for spatial surfaces using a multiple group rack.
Study finds minimal coloring numbers for torus links using rack colorings.
The paper introduces groupoid racks for spatial surfaces.
A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from th…
New algebraic structure helps distinguish braids.
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modu…
New rack and multiple group rack cohomology for surfaces in 3-sphere.
Study of generalized Legendrian racks and their GL-structures.
New method constructs multiple group racks, differing from known constructions.
A rack of order is a binary operation $\rack$ on a set of cardinality , such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all , and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all . …
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
Innovative rack theory applied to Legendrian links.
We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of …
We study Coxeter racks over and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
4-Legendrian permutation racks can't distinguish knots but recover classical invariants.
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 …
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 -quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, gen…
We introduce a modified rack algebra Z[X] for racks X with finite rack rank N. We use representations of Z[X] into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are stric…
A (t,s)-rack is a rack structure defined on a module over the ring . We identify necessary and sufficient conditions for two -racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t,s)-racks and give some computations an…
New algebraic structures help distinguish Legendrian knots.
This paper develops graph theory for racks and quasigroups.
We describe a presentation for the augmented fundamental rack of a link in the lens space . Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in . In this case, the counting rack invariants also include the informatio…
Survey of recent developments in racks and quandles.
Classifies good involutions in conjugation subquandles and racks.
Defines fundamental racks for braid spaces of complex reflection groups.
The paper constructs Yang-Baxter solutions using categorical augmented racks.
Paper proves Legendrian knots with same GL-rack have similar invariants.
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invari…
This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria…
New method to calculate 3-manifold invariants via skew-racks.
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
The study enumerates virtual quandles up to isomorphism.
We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander r…
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
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 that N -degenerate chains form a sub-complex of the classical complex defining rack…
Proposes a new notation for biracks to simplify rack structure analysis.
This paper extends rack and quandle covering theory using higher categorical Galois theory.
Analytic Lie rack structures on Leibniz algebras are characterized and rigid Lie algebras are identified.
We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Al…
Second part of a series on higher coverings of racks and quandles.
New definitions of rack and quandle modules are introduced, and shown to generalise the definitions previously studied by Andruskiewitsch, Etingof and Grana. This new construct is shown to coincide with Beck's general definition of a module in an arbitrary category. A theory of Abelian extensions of racks and quandles …
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.
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 sub-family of quandles satisfying the graphic axiom. For a large family of graphic qua…