arXiv research

This paper extends braid lifting to coloured braid groupoids for all simple disc covers.

Extends a formula for the homomorphism defect of a signature map to coloured braids.

We study a novel type of braid groups on a closed orientable surface $Σ$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $Σ$; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles ove…

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the …

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let π_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longit…

Extends knot invariant computation to symmetrically colored sl_N.

Formulates quantum jet bundles over noncommutative algebras with connections and braiding.

In this paper we look for closed expressions to calculate the number of colourings of prime knots for given linear Alexander quandles. For this purpose the colouring matrices are simplified to a triangular form, when possible. The operations used to perform this triangularization preserve the property that the entries …

We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-c…

A G-coloured knot is a knot together with a representation of its knot group onto G. Two G-coloured knots are said to be rho-equivalent if they are related by surgery around unit framed unknots in the kernels of their colourings. The induced local move is a G-coloured analogue of the crossing change. For certain famili…

For p=3 and for p=5 we prove that there are exactly p equivalence classes of p-coloured knots modulo (+/-1)--framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n left-hand (p,2)-torus knots with a given colouring when n=1,2,...,p. This gives a 3-c…

A 2-complex requires at least 12 colours to avoid edge conflicts.

The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed by several authors with respect to orientable 3-manifolds up to 28 coloured tetrahedra, non-orientable 3-manifolds up to 26 coloured tetrahedra, genus two 3-manifolds up to 34 coloured tetrahedr…

The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is…

Study of coloured invariants of torus knots using $\mathcal{W}$ algebras.

Colour vision has long fascinated scientists, who have sought to understand both the physiology of the mechanics of colour vision and the psychophysics of colour perception. We consider representations of colour in anatomically constrained convolutional deep neural networks. Following ideas from neuroscience, we classi…

We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of $U_q(\mathfrak{gl}(1|1))$. This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family o…

Characters from logarithmic VOAs linked to torus link invariants.

Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the in…

We prove that the reduced 2-coloured Khovanov homology detects the trefoil, using a spectral sequence to knot Floer homology.

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpre…

Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by L…

We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relati…

Grogan et al [11,12] have recently proposed a solution to colour transfer by minimising the Euclidean distance L2 between two probability density functions capturing the colour distributions of two images (palette and target). It was shown to be very competitive to alternative solutions based on Optimal Transport for c…

Motivated by a possible connection between the $\mathrm{SU}(N)$ instanton knot Floer homology of Kronheimer and Mrowka and $\mathfrak{sl}(N)$ Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…

The study of symmetries in manifolds derived from colored polytopes.

We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.

Hierarchical quandles extend diquandles and multi-quandles for link invariants.

The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also …

We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a $O(N^{\frac{3}{2} \mathrm{cw}} \mathrm{poly}(n))$ time algorithm to compute any Resh…

New geometric invariant from disc intersections captures all coloured Jones polynomials.

We present the construction of a large class of homogeneous KT, HKT and QKT manifolds, $G/K$, using an invariant metric on $G$ and the canonical connection. For this a decomposition of the Lie algebra of $G$ is employed, which is most easily described in terms of colourings of Dynkin diagrams of simple Lie algebras. KT…

Constructs universal link invariants from intersections in configuration spaces.

New approach to electric group for knots and links.

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In t…

We present a novel approach to adjust global image properties such as colour, saturation, and luminance using human-interpretable image enhancement curves, inspired by the Photoshop curves tool. Our method, dubbed neural CURve Layers (CURL), is designed as a multi-colour space neural retouching block trained jointly in…

We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be interpreted as a space of colourings of a diagram of the link $L$ with colours f…

A reliable human skin detection method that is adaptable to different human skin colours and illu- mination conditions is essential for better human skin segmentation. Even though different human skin colour detection solutions have been successfully applied, they are prone to false skin detection and are not able to c…

Categorifies Jones polynomial using Lie theory.

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for this invariants, showing that they ca…

This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the plane and the recent contributions by Wessel. Finally we will conclude with a discus…

This paper solves a problem in 3D geometry by defining a canonical partition for certain manifolds.

Single-image super-resolution (SISR) is a canonical problem with diverse applications. Leading methods like SRGAN produce images that contain various artifacts, such as high-frequency noise, hallucinated colours and shape distortions, which adversely affect the realism of the result. In this paper, we propose an altern…

Polynomial invariant derived from birack labelling of knots.

Unified quantum invariants via intersections of embedded Lagrangians.

New quantum knot invariants derived from Verma modules.