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.
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.
Study rack invariants for links in lens space .
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.
New algebraic structure helps distinguish braids.
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…
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…
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
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…
New rack invariants detect geometric properties of Legendrian knots.
New method constructs multiple group racks, differing from known constructions.
New rack and multiple group rack cohomology for surfaces in 3-sphere.
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…
The paper introduces a new coloring invariant for spatial surfaces using a multiple group rack.
New method to calculate 3-manifold invariants via skew-racks.
New algebraic structures help distinguish Legendrian knots.
4-Legendrian permutation racks can't distinguish knots but recover classical 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…
Innovative rack theory applied to Legendrian links.
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…
New shadow homotopy invariant defined for links.
Paper proves Legendrian knots with same GL-rack have similar invariants.
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…
Link invariants defined from finite crossed modules and Reidemeister pairs.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizi…
The paper introduces groupoid racks for spatial surfaces.
This paper develops graph theory for racks and quasigroups.
Analytic Lie rack structures on Leibniz algebras are characterized and rigid Lie algebras are identified.
We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to -reduced rack homology when the birack is a rack. We introduce augmente…
Study of generalized Legendrian racks and their GL-structures.
New racks defined; properties of rack representations explored.
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…
Paper constructs infinitely many pairs of Seifert surfaces for each link.
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
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 …
Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcompl…
Survey of recent developments in racks and quandles.
Classifies good involutions in conjugation subquandles and racks.
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bu…
Defines fundamental racks for braid spaces of complex reflection groups.
Invariants for long knots defined using algebraic and categorical methods.
The paper constructs Yang-Baxter solutions using categorical augmented racks.
The paper computes rack homology for a specific family of quandles.
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
We extend the rack algebra Z[X] defined by Andruskiewitsch and Grana to the case of biracks, enabling a notion of birack modules. We use these birack modules to define an enhancement of the birack counting invariant generalizing the birack module counting invariant in [8]. We provide examples demonstrating that the enh…