Algebraic Modeling of Topological and Computational Structures and Applications

The Yokonuma–Hecke algebras are naturally related to the framed braid group and they support a Markov trace. Consequently, invariants for various types of links (framed, classical, singular and transverse) are derived from these algebras. In this paper, we present results about these invariants and their properties. We focus, in particular, on the family of 2-variable classical link invariants that are not topologically equivalent to the HOMFLYPT polynomial and on the 3-variable classical link invariant that generalizes this family and the HOMFLYPT polynomial.

In this survey we collect all results regarding the construction of the Framization of the Temperley–Lieb algebra of type A as a quotient algebra of the Yokonuma–Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma–Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley–Lieb algebra. From the Framization of the Temperley–Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pairs of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.

We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma–Hecke algebras. The isomorphism relates affine Yokonuma–Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma–Hecke algebras, and in turn, to produce invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma–Hecke algebras. In general, the obtained invariants form an infinite family of 3-variable polynomials. As a consequence of the construction via the isomorphism, we reduce the number of invariants to study, given the number of connected components of a link. In particular, if the link is a classical link with N components, we show that N invariants generate the whole family.

We give a cross look to two framizations of the Hecke algebra of type \(\mathtt {B}\). One of these is a particular case of the cyclotomic Yokonuma–Hecke algebra. The other one was recently introduced by the author, J. Juyumaya and S. Lambropoulou. The purpose of this paper is to show the main concepts and results of both framizations, giving emphasis to the second one, and to provide a preliminary comparison of the invariants constructed from both framizations.

In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in \(F\times S^1\) and \(N\hat{\times }S^1\), where F is an orientable and N an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. Transitions between arrow diagrams and alternative diagrams are presented. We recall results about the Kauffman bracket and HOMFLYPT skein modules of some Seifert manifolds using arrow diagrams, namely lens spaces, a product of a disk with two holes times \(S^1\), \(\mathbb {R}P^3 \# \mathbb {R}P^3\), and prism manifolds. We also present new bases of the Kauffman bracket and HOMFLYPT skein modules of the solid torus and lens spaces.

In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces L(p, 1), \(\mathcal {S}\left( L(p,1) \right) \), via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, X, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between \(\mathcal {S}\left( L(p,1) \right) \) and \(\mathcal {S}(\mathrm{ST})\) is established in Diamantis et al. (J Knot Theory Ramif, 25:13, 2016, [5]) and it is shown that in order to compute \(\mathcal {S}\left( L(p,1) \right) \), it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of \(\mathcal {S}(\mathrm{ST})\), \(\varLambda \), presented in Diamantis and Lambropoulou (J Pure Appl Algebra, 220(2):577–605, 2016, [4]). The solution of this infinite system of equations is very technical and is the subject of a sequel work (Diamantis and Lambropoulou, The HOMFLYPT skein module of the lens spaces L(p, 1) via braids, in preparation, [2]).

We construct some Hecke-type algebras, and most notably the quotient algebra \(\mathrm {H}_{2,n}(q)\) of the group-algebra \({\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}\) of the mixed braid group \(B_{2,n}\) with two identity strands and n moving ones, over the quadratic relations of the classical Hecke algebra for the braiding generators. The groups \(B_{2,n}\) are known to be related to the knot theory of certain families of 3-manifolds, and the algebras \(\mathrm {H}_{2,n}(q)\) are aimed for the construction of invariants of oriented knots and links in these manifolds. To this end, one needs a suitable basis of \(\mathrm {H}_{2,n}(q)\), and we have singled out a subset \(\varLambda _n\) of this algebra for which we proved it is a spanning set, whereas ongoing research aims at proving it to be a basis.

In this paper we study the kernel of the homomorphism \(B_{g,n} \rightarrow B_n\) of the braid group \(B_{g,n}\) in the handlebody \(\mathscr {H}_g\) to the braid group \(B_n\). We prove that this kernel is semi-direct product of free groups. Also, we introduce an algebra \(H_{g,n}(q)\), which is some analog of the Hecke algebra \(H_n(q)\), constructed by the braid group \(B_n\).

In this lecture note we survey results obtained under the research program Thalis (Kechagias, J. Homotopy Relat. Struct. 8(2), 201–229, (2013), [24], J. Pure Appl. Algebra 219(4), 839–863, (2015), [25]) and place them in the context of algebraic topology. It is divided into two parts. In the first part, we survey infinite loop spaces, \(\varOmega \)-spectra, and their relation with the symmetric groups. In the second part, we express the component Dyer–Lashof coalgebras as subalgebras of a cofree unstable coalgebra on two cogenerators using an extension of the Peterson conjecture. We also compare and approximate \(H^{*}\left( Q_{0}S^{0};\mathbb {Z}/p \mathbb {Z}\right) \) with certain free objects using modular invariants. A new basis for \(H^{*}\left( B\varSigma _{\infty }; \mathbb {Z}/p \mathbb {Z}\right) \) is provided.

With a focus on one-dimensional periodic boundary systems, we describe the application of extensions of the Gauss linking number of closed rings to open chains and, then, to systems of such chains via the periodic linking and periodic self-linking of chains. These lead to the periodic linking matrix and its associated eigenvalues providing measures of entanglement that can be applied to complex systems. We describe the general one-dimensional case and applications to one-dimensional Olympic gels and to tubular filamental structures.

Knotoid diagrams are defined in analogy to open ended knot diagrams with two distinct endpoints that can be located in any region of the diagram. The height of a knotoid is the minimal crossing distance between the endpoints taken over all equivalent knotoid diagrams. We define two knotoid invariants; the affine index polynomial and the arrow polynomial that were originally defined as virtual knot invariants given in (Kauffman, J Knot Theory Ramif 21(3), 37, 2012) [6], (Kauffman, J Knot Theory Ramif 22(4), 30, 2013) [8], respectively, but here are described entirely in terms of knotoids in \(S^2\). We reprise here our results given in (Gügümcü, Kauffman, Eur J Combin 65C, 186–229, 2017) [3] that show that both polynomials give a lower bound for the height of knotoids.

By the closure operation, knots can be represented by cyclic braids, which can be unfolded as periodic complex valued functions. Their description by Fourier series allows an approximation by finite Laurent polynomials g(z). We define an algebraic discriminant \(\varDelta ^n_g(z)\), such that an n-braid is given by those g(z) satisfying the condition (S) of having all roots not on the unit circle. We study property (S) from the algebraic and topological viewpoint. Using further algebraic conditions for g(z) we obtain algebraic representations of cyclic braids in thickened surfaces, which represent periodic boundary conditions.

Ionic Liquids (ILs) are organic salts with melting temperatures below \({100^\circ }\) C. They are characterized by an exceptional combination of properties that renders them very good candidates for use in many cutting-edge technological applications. The organic and simultaneously ionic nature of the constitutive ions results in diverse interactions that directly affect the microscopic structure and the dynamical behaviour of ILs. Molecular simulation methods using optimized force fields are applied for the study of the complex dynamics and the spatial organization in ILs.

In this paper, we extend the formal definition of topological surgery by introducing new notions in order to model natural phenomena exhibiting it. On the one hand, the common features of the presented natural processes are captured by our schematic models and, on the other hand, our new definitions provide the theoretical setting for examining the topological changes involved in these processes.

This is a survey article concerning the groups of automorphisms of curves defined over algebraically closed fields of positive characteristic, their representations and applications to their deformation theory.

This paper constitutes a first attempt at constructing semantic theories over institutions and examining the logical relations holding between different such theories. Our results show that this approach can be very useful for theoretical computer science (and may also contribute to the current philosophical debate regarding the semantic and the syntactic presentation of scientific theories). First we provide a definition of semantic theories in the institution theory framework - in terms of a set of models satisfying a given set of sentences - using the language-independent satisfaction relation characterizing institutions (Definition 17.3). Then we give a proof of the logical equivalence holding between the syntactic and the semantic presentation of a theory, based on the Galois connection holding between sentences and models (Theorem 17.1). We also show how to integrate and combine semantic theories using colimits (Theorem 17.2). Finally we establish when the output of a model-based software verification method applied to a semantic theory over an institution also holds for a semantic theory defined over a different institution (Theorem 17.3).

We collect results related to generic constructions and generic limits for semantic and syntactic cases. It is considered both by pure model theory approach and by the institutional approach.

We present a verification framework developed by researchers of the National Technical University of Athens as part of the Research Project Thalis “Algebraic Modeling of Topological and Computational Structures and Applications”. The proposed framework combines two different specification and theorem-proving systems, in order to facilitate the modeling and analysis of critical software systems. On the one hand, the CafeOBJ algebraic specification language offers executable, composable specifications, and insightful information about the proofs of desired invariant properties. On the other hand, Athena, an interactive theorem-proving system, provides automation and soundness guarantees for its results, as well as detailed structured proofs. Although having conducted complicated case studies (references to which are provided in the paper), here we focus on explaining the steps of the proposed hybrid methodology as clearly as possible, through an illustrative example of a simple mutual exclusion protocol.

The constantly augmenting loads of the aviation industry inevitably define the evolutions in the field of Air Traffic Control. Relevant regulations are changing, in order to accommodate the increase in passenger, flight, and cargo numbers. This paper presents the design of an innovative rule base for the Air Traffic Control regulations during the take-off and landing phases, covering both current and future separation standards of ICAO and FAA. The rule base consists of the rules implementing the Air Traffic Control regulations, and a database containing characteristics of airports and aircraft. The proposed rule base constitutes a flexible tool for the computation of the aircraft separation according to current and future regulations, useful in the fields of conflict detection, conflict avoidance, and scheduling aircraft landings. A further application will be as a decision support tool in real-time environments, guaranteeing the enforcement of all the separation standards.

Affine contours may be viewed as an abstraction of the notion of musical intervals and are closely related to sequential machines. We show that every commutative affine musical contour actually simulates the classical one \(c:\mathbb {Z}_{12} \times \mathbb {Z}_{12} \rightarrow \mathbb {Z}_{12}\), \(c(s,t)=t-s\).

A hyperoperation on the pixels of a two-dimensional picture is introduced and studied. In this setup pictures are defined as a specific type of rectangular graphs and the picture hyperoperation is given by virtue of the notion of the path inside such a picture. Using this hyperoperation objects can naturally be defined inside pictures in an algebraic way and this concept can be utilized in order to formally represent and compare video content using algebraic semiotics.

The purpose of this article is to present scientific research results of Karali et al (Proceedings of 8th International Conference on Bioinformatics and Bioengineering (BIBE), 2008, [14]), Karali et al (Inf Technol Biomed, 15(13):381–6, 2011, [15]), Karali et al (J Biosci Med (JBM), 1:6–9, 2013, [16]), Karali et al (Int J Comput Vis, 321–331, 1988 [24]) concerning medical imaging, especially in the fields of image reconstruction in Emission Tomography and image segmentation. Image reconstruction in Positron Emission Tomography (PET) uses the collected projection data of the object/patient under examination. Iterative image reconstruction algorithms have been proposed as an alternative to conventional analytical methods. Despite their computational complexity, they become more and more popular, mostly because they can produce images with better contrast-to-noise (CNR) and signal-to-noise (SNR) ratios at a given spatial resolution, compared to analytical techniques. In Sect. 23.1 of this study we present a new iterative algorithm for medical image reconstruction, under the name Image Space Weighted Least Squares (ISWLS) (Karali et al, Proceedings of 8th International Conference on Bioinformatics and Bioengineering (BIBE), 2008, [14]). In (Karali et al, Proceedings of 8th International Conference on Bioinformatics and Bioengineering (BIBE), 2008, [14]) we used phantom data from a prototype small-animal PET system and the methods presented are applied to 2D sinograms. Further, we assessed the performance of the new algorithm by comparing it to the simultaneous versions of known algorithms (EM-ML, ISRA and WLS). All algorithms were compared in terms of cross-correlation coefficient, reconstruction time and CNRs. ISWLS have ISRA’s properties in noise manipulation and WLS’s acceleration of reconstruction process. As it turned out, ISWLS presents higher CNRs than EM-ML and ISRA for objects of different sizes. Indeed ISWLS shows similar performance to WLS during the first iterations but it has better noise manipulation. Section 23.5 of this study deals with another important field of medical imaging, the image segmentation and in particular the subject of deformable models. Deformable models are widely used segmentation methods with scientifically accepted results. In Karali et al (Int J Comput Vis, 321–331, 1988 [24]) various methods of deformable models are compared, namely the classical snake (Kass et al, Int J Comput Vis, 321–331, 1988, [25]), the gradient vector field snake (GVF snake) (Xu, IEEE Proceedings on Computer Society Conference on Computer Vision and Pattern Recognition, 1997, [36]) and the topology-adaptive snake (t-snake) (Mcinerney, Topologically Adaptable Deformable Models For medical Image Analysis, 1997, [29]), as well as the method of self-affine mapping system (Ida and Sambonsugi, IEEE Trans Imag Process, 9(11), 2000 [22]) as an alternative to snake models. In Karali et al (Int J Comput Vis, 321–331, 1988 [24]) we modified the self-affine mapping system algorithm as far as the optimization criterion is concerned. The new version of self-affine mapping system is more suitable for weak edges detection. All methods were applied to glaucomatic retinal images with the purpose of segmenting the optical disk. The methods were compared in terms of segmentation accuracy and speed. Segmentation accuracy is derived from normalized mean square error between real and algorithm extracted contours. Speed is measured by algorithm segmentation time. The classical snake, T-snake and the self-affine mapping system converge quickly on the optic disk boundary comparing to GVF-snake. Moreover the self-affine mapping system presents the smallest normalized mean square error (nmse). As a result, the method of self-affine mapping system presents adequate segmentation time and segmentation accuracy, and significant independence from initialization.