Related papers: Isotopy classes of 3-periodic net embeddings
The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by…
We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes…
We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat \beta$ around the core of the corresponding…
This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications.Periodic structures such as solid crystalline materials or textiles were previously classified in…
For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the…
The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap…
Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots…
We will construct differential forms on the embedding spaces Emb(R^j,R^n) for n-j>=2 using configuration space integral associated with 1-loop graphs, and show that some linear combinations of these forms are closed in some dimensions.…
The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time,…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
A Newton graph of order $r( \geqslant 2)$ is a cellularly embedded toroidal graph on $r$ vertices, $2r$ edges and $r$ faces that fulfils certain combinatorial properties (Euler, Hall). The significance of these graphs relies on their role…
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each element of $\mathcal{E}$ and an edge for each…
We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…
We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number,…
This work deals with the scattering entropy of quantum graphs in many different circumstances. We first consider the case of the Shannon entropy and then the R\'enyi and Tsallis entropies, which are more adequate to study distinct…
A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.
A weave is the lift to the Euclidean thickened plane of a set of infinitely many planar crossed geodesics, that can be characterized by a number of sets of threads describing the organization of the non-intersecting curves, together with a…
We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph…
A {\em net} is a graph consisting of a triangle $C$ and three more vertices, each of degree one and with its neighbour in $C$, and all adjacent to different vertices of $C$. We give a polynomial-time algorithm to test whether an input graph…