arXiv research

The paper establishes isomorphisms and constructs colored versions of Lawrence representations.

Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can…

The paper introduces two-tone colorings for links and shows conditions for surjective dihedral representations.

We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.

The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.

This paper shows all elements in the 3-colorable subgroup of Thompson's group give 3-colorable links.

We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantizatio…

New $p$-colorable subgroup derived from Thompson's group.

For any Legendrian knot $K$ in standard contact ${\mathbb R}^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce an ungraded $n$-colored ruling polynomial, $R^1_{n,K}(q)$…

We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum repr…

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…

We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman homologies carrying the symmetric tensor products of the vector representation for th…

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 knot representations and spherical quandle colorings.

Study on distinguishing mutant knots using specific representations.

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 $\fsl_3$, …

In this paper we investigate the asymptotic behavior of the colored HOMFLY polynomial of the figure eight knot associated with the symmetric representation. We establish an analogous asymptotic expansion for the colored HOMFLY polynomial. From the asymptotic behavior we show that the Chern-Simons invariants and twisted…

In this paper, we show that a simple coloring scheme can improve, both theoretically and empirically, the expressive power of Message Passing Neural Networks(MPNNs). More specifically, we introduce a graph neural network called Colored Local Iterative Procedure (CLIP) that uses colors to disambiguate identical node att…

New computations show sl(N) homology is related to SU(N) representations of knots.

We prove that the Witten--Reshetikhin--Turaev $\mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposa…

Moduli space linked to Tait colorings of planar graphs.

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point con…

Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten mod…

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 use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress t…

We establish relationships between two classes of invariants of Legendrian knots in $\mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $β\subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded …

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…

Novel symmetry found in colored HOMFLY polynomials from superalgebras.

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.

We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl(N) link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl(N) link homologies. By studying the underlying deformed higher r…

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the …

Formula for colored Links-Gould polynomial with genus bounds.

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 $2$. Our conjecture is motivated by a structure theorem for the degree …

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…

We show that any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof, Rowell and Witherspoon showing that the braid group images are finite. In particular, our result answers their question regarding finitene…

Study on coloring virtual tangles with integer and modular arithmetic.

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…

In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge…

We generalize the colored Jones polynomial to $4$-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.

Study on the growth of colored Jones polynomial for figure-eight knot cables.

We provide methods to compute the colored HOMFLY polynomials of knots and links with symmetric representations based on the linear skein theory. By using diagrammatic calculations, several formulae for the colored HOMFLY polynomials are obtained. As an application, we calculate some examples for hyperbolic knots and li…

In this paper we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ f…

We first study superpolynomial associated to triply-graded reduced colored HOMFLY-PT homology. We propose conjectures of congruent relations and cyclotomic expansion for it. We prove conjecture of $N=1$ for torus knot case, through which we obtain the corresponding invariant $α(T(m,n))=-(m-1)(n-1)/2$. This is closely r…

We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, …

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…