Related papers: Peripheral elements in reduced Alexander modules
Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain…
We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander…
We consider the problem of how many components to retain in the application of principal component analysis when the dimension is much higher than the number of observations. To estimate the number of components, we propose to sequentially…
Many real-world networks exhibit a multicores-periphery structure, with densely connected vertices in multiple cores surrounded by a general periphery of sparsely connected vertices. Identification of the multicores-periphery structure can…
For a horizontally linked module, over a commutative semiperfect Noetherian ring $R$, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. It is shown that under certain conditions the depth of a…
Walter Neumann showed that the topology of a ``regular'' algebraic curve V in C^2 is determined up to proper isotopy by some link in S^3 called the link at infinity of V. In this note, we compute the Alexander module over C[t^{\pm 1}] of…
The Levine-Tristram signature admits an n-variable extension for n-component links: it was first defined as an integer valued function on $(S^1\setminus\{1\})^n$, and recently extended to the full torus $T^n$. The aim of the present article…
In recent work, comonads and associated structures have been used to analyse a range of important notions in finite model theory, descriptive complexity and combinatorics. We extend this analysis to Hybrid logic, a widely-studied extension…
The wholeness, conceived and developed by Christopher Alexander, is what exists to some degree or other in space and matter, and can be described by precise mathematical language. However, it remains somehow mysterious and elusive, and…
As a previous result, it has shown that every sphere-link consisting of trivial components is a ribbon sphere-link. In this note, it is shown that for every closed oriented disconnected surface F with just one non-sphere component, every…
It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular $2x2$ matrices. In…
Bar-Natan's Zh-construction associates to each $n$ component virtual link diagram $L$ an $(n+1)$ component virtual link diagram $Zh(L)$. If $L_0,L_1$ are equivalent virtual link diagrams, then $Zh(L_0),Zh(L_1)$ are equivalent as semi-welded…
The study of the sub-structure of complex networks is of major importance to relate topology and functionality. Many efforts have been devoted to the analysis of the modular structure of networks using the quality function known as…
We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot and the fundamental quandles/groups of…
Some networked systems can be better modelled by multilayer structure where the individual nodes develop relationships in multiple layers. Multilayer networks with similar nodes across layers are also known as multiplex networks. This…
Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is…
We introduce and study a nontrivial generalization of uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right (resp., left) weakly uniserial ring is a ring which…
We define a sequence of integer-valued invariants $\gamma^k(L)$ for a $3$-component link $L$. We prove that the resulting $\gamma$-invariants are invariant under concordance, and more generally under weak cobordism, and that they lift…
Let $f:\CN \rightarrow \C $ be a polynomial, which is transversal (or regular) at infinity. Let $\U=\CN\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give…
We revisit generic vanishing results for perverse sheaves with any field coefficients on a complex semi-abelian variety, and indicate several topological applications. In particular, we obtain finiteness properties for the integral…