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 on knots using 17 colors, finding specific color assignments.
The paper discusses knot colorings and their invariants using Goeritz matrices.
The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.
We prove that any -colorable knot is presented by an -colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially -colored diagrams of the knot. We also prove a similar result for any -colorable ribbon -knot.
New colored knot Floer homology defined using infinite full twists.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
New knot homology invariant grows exponentially with color.
An invariant for knots with colored bonds respects HOMFLYPT relation.
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…
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…
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each typ…
The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.
The paper extends surface link coloring theory to triplane diagrams and knots.
Clarifies how knot homset invariants relate to diagram colorings.
The paper constructs Goeritz matrices from Dehn colorings.
The paper connects GL-racks to knot coloring invariants.
We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial. Along the way we also calculate the Kauffman bracket skein module of all 2-brid…
Proved colored HOMFLY-PT polynomials for specific knots.
This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purpo…
Generalizes region select game to -colored knot diagrams.
This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false thet…
New R-equivalence classes found for torus knot diagrams.
An algorithm determines knot colorability and determinants from petal projections.
Improved lower bound for knot coloring using quandles.
Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.
Enhances psyquandle invariants for singular and pseudoknots.
Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that o…
Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter-Silver-Williams clarifies the relationship between the two focusing on how one transitions between Fox and Dehn colorings. I…
We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differentials and of higher degree than the standard homology differentials, and give access to typically more complex homology groups.
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…
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…
New formula recovers degree of colored Jones polynomials for pretzel knots.
Study on the growth of colored Jones polynomial for figure-eight knot cables.
Unified ADO and colored Jones polynomials for knots.
New -colorable subgroup derived from Thompson's group.
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
Researchers compute and predict knot volumes using colored Jones polynomials.
The tail of the colored Jones polynomial of an alternating link is a -series invariant whose first terms coincide with the first terms of the -th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information.
Khovanov homology detects essential surfaces in knot complements.
We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that relates the asymptotic behavior of the colored Jones polynomial of a knot to the…
In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on …
The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.
Researchers create functors to match colored homologies of knots and links.
In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the -th colored Jones polynomials evaluated at -th root of unity with a fixed limiting ratio, , of and . We find out the…
Invariants for colored links found, with topological protection of certain knots.
The Penrose-Kauffman polynomial connects knot theory to graph coloring.
New homologies prove -holonomicity of knot polynomials.