Hamid Abchir’s research while affiliated with University of Hassan II Casablanca and other places

We assign a new polynomial to any vertex-signed checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Jones-Kauffman polynomial for checkerboard-colorable virtual links.

1 We give a non-left-orderability criterion for involutory quandles of non-split links. We use this criterion to show that the involutory quandle of any non-trivial alternating link is not left-orderable, thus improving Theorem 8.1. proven by Raundal et al. (Proceedings of the Edinburgh Mathematical Society (2021 64), page 646). We also use the criterion to show that the involutory quandles of augmented alternating links are not left-orderable. We introduce a new family of links containing all non-alternating and quasi-alternating 3-braid closures and show that their involutory quandles are not left-orderable. This leads us to conjecture that the involutory quandle of any quasi-alternating link is not left-orderable.

We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual links.

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(−2, 3, 2l + 1), l ≥ 0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more than one component.

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component.

For each connected alternating tangle, we provide an infinite family of non-left-orderable L-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational homology 3-sphere is an L-space if and only if it is non-left-orderable. These 3-manifolds are obtained as Dehn fillings of the double branched covering of any alternating encircled tangle. We give a presentation of these non-left-orderable L-spaces as double branched coverings of S3, branched over some specified links that turn out to be hyperbolic. We show that the obtained families include many non-Seifert fibered spaces. We also show that these families include many Seifert fibered spaces and give a surgery description for some of them. In the process we give another way to prove that the torus knots T(2,2m+1) are L-space-knots as has already been shown by Ozsváth and Szabó in [24].

k$-Para-K\"ahler Lie algebras are a generalization of para-K\"ahler Lie algebras$(k=1)$and constitute a subclass of$k$-symplectic Lie algebras. In this paper, we show that the characterization of para-K\"ahler Lie algebras as left symmetric bialgebras can be generalized to$k$-para-K\"ahler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized$S$-matrices. We determine then all the$k$-symplectic Lie algebras of dimension$(k+1)$ and all the six dimensional 2-para-K\"ahler Lie algebras.

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.

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of the Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (01), 2015) is false.

A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any x, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, $x\rhd y=y+\sum_{n=1}^\infty A_{n,1}(x,\ldots,x,y),$where $A_{n,1}:V\times\ldots\times V\Leftarrow V$ is an n+1-multilinear map symmetric in the n first arguments. In this case, $A_{1,1}$ is exactly the left Leibniz product associated to $\rhd$. Any left Leibniz algebra $(\mathfrak{h},[\;,\;])$ has a canonical analytic linear Lie rack structure given by $x\stackrel{c}{\rhd} y=\exp(\mathrm{ad}_x)(y)$, where $\mathrm{ad}_x(y)=[x,y]$. In this paper, we show that a sequence $(A_{n,1})_{n\geq1}$ of n+1-multilinear maps on a vector space V defines an analytic linear Lie rack structure if and only if $[\;,\;]:=A_{1,1}$ is a left Leibniz bracket, the $A_{n,1}$ are invariant for $(V,[\;,\;]:)$ and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra $(V,[\;,\;])$ are trivial. On the other hand, given a left Leibniz algebra $(\mathfrak{h},[\;,\;])$, we show that there is a large class of (analytic) linear Lie rack structures on $(\mathfrak{h},[\;,\;])$ which can be built from the canonical one and invariant multilinear symmetric maps on $\mathfrak{h}$. A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that $\mathfrak{sl}_2(\mathbb{R})$ and $\mathfrak{so}(3)$ are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra.