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 of quandle coloring quivers with dihedral quandles.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
The study characterizes torus links' coloring quivers using dihedral quandles.
New polynomial invariants from quandle action quivers.
In-degree quiver polynomials for surface-links computed.
Enhanced invariant for linkoids using quivers.
Paper studies knotoid chirality using shadow quandle colorings and invariants.
New polynomial invariants for knots and links from quandle coloring quiver decategorification.
Enhances psyquandle invariants for singular and pseudoknots.
We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e., giving the set of quandle colorings the structure of a small…
We incorporate quandle cocycle information into the quandle coloring quivers we defined in arXiv:1807.10465 to define weighted directed graph-valued invariants of oriented links we call \textit{quandle cocycle quivers}. This construction turns the quandle cocycle invariant into a small category, yielding a categorifica…
Enhances knot and link invariants using quandle modules.
We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
New invariants for singular knots and links defined using shadow structures.
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…
For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic repre…
In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces…
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…
New invariants defined for knots and links using quandle representations.
As one of the problems in his list [20], T. Ohtsuki proposed to study relations between quandle cocycle invariants and quantum invariants. The aim of this paper is to answer one of those questions. We prove that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant defined by $(\Z…
New colored link invariants using multi-quandles.
When a boudnary-parabolic representation of a link group to PSL(2,) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related wit…
Symmetric quandles provide new insights into link colorings.
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
New link colorings using quandle rings and idempotents are stronger than existing methods.
In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial is vanishing, then admits a non-trivial coloring by any non-trivial Alexander quandle , and that if , then admits only the trivial coloring by any Alexa…
Paper provides criteria to detect non-admissible quandles via coloring.
This paper studies quandles with one non-trivial column and their properties.
If a knot has the Alexander polynomial not equal to 1, then it is linear -colorable. By means of such a coloring, such a knot is given an upper bound for the minimal quandle order, i.e., the minimal order of a quandle with which the knot is quandle colorable. For twist knots, we study the minimal quandle orders in d…
We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of…
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle invariants of composite knots. We investigate t…
New knot invariants derived from biquandle quivers.
Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
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…
Explains biquandle brackets and quivers for a topology talk.
New categorifications of biquandle brackets defined.
New algebraic structures help categorify link invariants.
We define a functor from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle , there is a one-to-one correspondence between the set of -colorings and that of -colorings diagrammatically for any …
New R-equivalence classes found for torus knot diagrams.
The paper connects knot representations and spherical quandle colorings.
Polynomial invariant of quandles counts random link colorings.
Improved lower bound for knot coloring using quandles.
Study satellite knots and their quandles related to incompressible tori.
The paper explores knotoid invariants on the plane using quandles and cocycles.
Enhanced symplectic quandle colorings detect causal structure in spacetime diagrams.
We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun by the second author et al., finding biquandle a…
The number of colorings of a knot diagram by a quandle has been shown to be a knot invariant by CJKLS using quandle cohomology methods. In a previous paper by the second named author, the CJKLS invariant was refined and, in particular, it was shown that the number of colorings is an invariant directly without resorting…