Papers by Mohamed Elhamdadi
Cornell University - arXiv, May 22, 2020
We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colo... more We investigate Fox colorings of knots that are 17-colorable. Precisely, we prove that any 17-colorable knot has a diagram such that exactly 6 among the seventeen colors are assigned to the arcs of the diagram.
Springer Proceedings in Mathematics & Statistics, 2020
We use the structural aspects of the f-quandle theory to classify, up to isomorphisms, all f-quan... more We use the structural aspects of the f-quandle theory to classify, up to isomorphisms, all f-quandles of order n. The classification is based on an effective algorithm that generate and check all f-quandles for a given order. We also include a pseudocode of the algorithm.
Communications of The Korean Mathematical Society, May 15, 2019
We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, m... more We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.
This article gives a foundational account of various characterizations of framed links in the 3-s... more This article gives a foundational account of various characterizations of framed links in the 3-sphere.
Twisted-Racks, Twisted-Quandles, their Extensions and Cohomology
arXiv: Rings and Algebras, 2016
The purpose of this paper is to introduce and study the notions of twisted rack and twisted-quand... more The purpose of this paper is to introduce and study the notions of twisted rack and twisted-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order twisted-quandles. Moreover, we define modules over twisted-racks, discuss extensions and define a cohomology complex for twisted quandles.
This article gives the foundations of the colored Jones polynomial for singular knots. We extend ... more 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 theta function identity that goes back to Ramanujan.
The study of singular knots, or equivalently rigid 4-valent graphs, and their invariants was gene... more The study of singular knots, or equivalently rigid 4-valent graphs, and their invariants was generated largely by the theory of Vassiliev invariants. Many existing knot invariants have been extended to singular knot invariants. In [3], Birman introduced braids in the theory of Vassiliev via the singular braids and conjectured that the monoid of singular braids maps injectively into the group algebra of the braid group. A proof of this conjecture was given by Paris in [25]. Fiedler extended the Kauffman state models of the Jones and Alexander polynomials to the context of singular knots [6]. In [7] Gemein investigated extensions of the Artin representation and the Burau representation to the singular braid monoid and the relations between them. Juyumaya and Lambropoulou constructed a Jones-type invariant for singular links using a Markov trace on a variation of the Hecke algebra [15]. In [20] Kauffman and Vogel defined a polynomial invariant of embedded 4-valent graph in R3 extending...
Journal of Knot Theory and Its Ramifications, 2021
We generalize the notion of the quandle polynomial to the case of singquandles. We show that the ... more We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.
Mediterranean Journal of Mathematics, 2021
We extend the quandle cocycle invariant to oriented singular knots and links using algebraic stru... more We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called oriented singquandles and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.
Journal of Knot Theory and Its Ramifications, 2018
This is an Erratum for our paper “Singular Knots and Involutive Quandles” in [1]. It replaces Sec... more This is an Erratum for our paper “Singular Knots and Involutive Quandles” in [1]. It replaces Sec. 5 in the published version of the paper.
On the quantumdynamics of measurement with geometric algebra
We define self-distributive structures in the categories of coalgebras and cocommutative coalgebr... more We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The selfdistributive operations of these structures provide solutions of the Yang-Baxter equation, and, conversely, solutions of the Yang-Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All of the work here is informed via diagrammatic computations.
Symmetry, 2021
Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a... more Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural symmetries stabilised by formation of intrachain contacts. Here, we adapt and apply the algebraic structure of quandles to classify and distinguish chain topologies within the framework of circuit topology. We systematically study the basic circuit topology motifs and define quandle/bondle coloring for them. Next, we explore the implications of circuit topology operations that enable building complex topologies from basic motifs for the quandle coloring approach.
International Journal of Algebra and Computation
The purpose of this paper is to introduce and investigate the notion of derivation for quandle al... more The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions.
Journal of Knot Theory and Its Ramifications, 2022
In this short survey, we review recent results dealing with algebraic structures (quandles, psyqu... more In this short survey, we review recent results dealing with algebraic structures (quandles, psyquandles, and singquandles) related to singular knot theory. We first explore the singquandles counting invariant and then consider several recent enhancements to this invariant. These enhancements include a singquandle cocycle invariant and several polynomial invariants of singular knots obtained from the singquandle structure. We then explore psyquandles which can be thought of as generalizations of oriented singquandles, and review recent developments regarding invariants of singular knots obtained from psyquandles.
We generalize the colored Jones polynomial to 4-valent graphs. This generalization is given as a ... more 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.
The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Ho... more The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give a procedure to provide deformations of alternative algebras (resp. Malcev algebras) into Hom-alternative algebras (resp. Hom-Malcev algebras). Then it is used to supply examples for which we compute some cohomology invariants.
Journal of Mathematical Physics, 2021
Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain inte... more Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain interactions that "glue" two or more regions of a chain. The resulting fold topology is widely believed to be a determinant of biomolecular properties and function. Recently, knot theory has been extended to describe the topology of folded linear chains such as proteins and nucleic acids. To classify and distinguish chain topologies, algebraic structure of quandles has been adapted and applied. However, the approach is limited as apparently distinct topologies may end up having the same number of colorings. Here, we enhance the resolving power of the quandle coloring approach by introducing Boltzmann weights. We demonstrate that the enhanced coloring invariants can distinguish fold topologies with an improved resolution.
We investigate generalized derivations of n-BiHom-Lie algebras. We introduce and study properties... more We investigate generalized derivations of n-BiHom-Lie algebras. We introduce and study properties of derivations, \(( \alpha ^{s},\beta ^{r}) \)-derivations and generalized derivations. We also study quasiderivations of n-BiHom-Lie algebras. Generalized derivations of \((n+1)\)-BiHom-Lie algebras induced by n-BiHom-Lie algebras are also considered.
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Papers by Mohamed Elhamdadi