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.
Clarifies how knot homset invariants relate to diagram colorings.
Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister, and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant and number of colorings, and how these can be computed from either the coloring matrix or the Goeritz m…
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzb…
The paper connects GL-racks to knot coloring invariants.
We formulate a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra, and verify it for all torus knots and all simple Lie algebras of rank . Our conjecture is motivated by a structure theorem for the degree …
New weight systems derived from a specific Lie algebra for knot invariants.
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 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…
New derivation of knot invariants from universal invariant.
The paper constructs Goeritz matrices from Dehn colorings.
Improved lower bound for knot coloring using quandles.
New -colorable subgroup derived from Thompson's group.
This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective …
We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use twists, and, rather than bisecting (), cut the strip into sections. We call the resulting collection of t…
The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and …
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.
Unified ADO and colored Jones polynomials for knots.
We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, one obtains the relations from the knot group, and from the core group. Using the ternary o…
Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the de…
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …
Paper provides criteria to detect non-admissible quandles via coloring.
Extends knot invariant computation to symmetrically colored sl_N.
In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes c…
Researchers lift knot coloring polynomial to Habiro ring.
Study satellite knots and their quandles related to incompressible tori.
New knot polynomials derived from Nichols algebras and braided Hopf algebras.
Formula for colored invariants of torus knots linked to algebras.
The SL_3 colored Jones polynomial of the trefoil knot is a -holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this -holonomic sequence as a case study. On the one hand, our results are new and useful to q…
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwi…
We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular we show that the in…
Study on knots using 17 colors, finding specific color assignments.
We study singularities of algebraic curves associated with 3d N=2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T_K labeled by knots, whose partition functions package Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the defining equation, call…
This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in the sense that if a diagram of the knot under study admits such a coloring, then …
Unified theories for colored sl(2) knot homology.
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.
The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along a braid model of the colored Jones polynomial that was refined by Armond from th…
New colored knot Floer homology defined using infinite full twists.
Equivalent categories of groups and risandles for 3-manifolds.
Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for and clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…
In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-z…
We introduce several algebraic structures related to handlebody-knots, including -families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in -oriented spatial trivalent graph diagrams representing -oriented handlebody-k…
Study on quandle coloring quivers for (p, 2)-torus knots and links.
The colored Jones polynomial is a -polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A -series called a tail is obtained as the limit of the colored Jones polynomials for some link , for example, an alternating link. For the $\mathf…
We use a link invariant defined by Cimasoni-Florens to compute ρ-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that the twist knots of algebraic order two are linearly independent in the topological …
Study connects knot contact homology to Chern-Simons theory's large N limit.
New knot homology invariant grows exponentially with color.
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…