Narrow Results

An anti-associative algebra is a nonassociative algebra whose multiplication satisfies the identity $a(bc)+(ab)c=0.$ Such algebras are nilpotent. We describe the free anti-associativealgebras with a finite number of generators. Other types of nonassociative algebras, obtained either by the polarization process such as Jacobi–Jordan algebras, or obtained by deformation quantization, are associated with this class of algebras. Following Markl-Remm’s work [M. Markl and E. Remm, (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities, J. Homotopy Relat. Struct.10 (2015) 939–969], we describe the operads associated with these algebra classes and in particular the cohomology complexes related to deformations.

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using diagrams in ${ℝ}^3$.

Feller et al. showed the homology and the intersection form of a closed trisected four-manifold are described in terms of trisection diagram. In this paper, it is confirmed that we are able to calculate those of a trisected four-manifold with boundary in a similar way. Moreover, we describe a representative of the second Stiefel–Whitney class by the relative trisection diagram.

C.F. Gauss gave a necessary condition for a word to be the intersection word of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. M. Dehn provided a solution to the planarity problem [3] and subsequently, different solutions have been given by a number of authors (see [9]). However, all of these solutions are algorithmic in nature. As B. Grünbaum remarked in [7], “they are of the same aesthetically unpleasing character as MacLane’s [1937] criterion for planarity of graphs. A characterization of Gauss codes in the spirit of the Kuratowski criterion for planarity of graphs is still missing”. In this paper we use the work of J. Scott Carter [2] to give a necessary and sufficient condition for planarity of signed Gauss words which is analogous to Gauss’s original condition.

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

Quandles and their homologies are used to construct invariants of oriented links or oriented surface-links in 4-space. On the other hand the knot quandle can still be defined in the case where the links or surface-links are not oriented, but in this case it cannot be used to construct homological invariants. Here we introduce the notion of a quandle with a good involution, and its homology groups. We can use them to construct invariants of unoriented links and unoriented, or non-orientable, surface-links in 4-space.