Related papers: Biquasiles and Dual Graph Diagrams
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram.…
We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal…
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain…
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric…
In 2014 Andrey Perfiliev introduced the so-called electric invariant for non-oriented knots. This invariant was motivated by using Kirchhoff's laws for the dual graph of the knot diagram. Later, in 2020, Anastasiya Galkina generalised this…
We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known…
Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words…
We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the…
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the…
We introduce \textit{Niebrzydowski algebras}, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and…
Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph…
We introduce a new infinite family of enhancements of the biquandle homset invariant called biquandle arrow weights. These invariants assign weights in an abelian group to intersections of arrows in a Gauss diagram representing a classical…
Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects…
Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical…
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.
Whiteley \cite{wh} gives a complete characterization of the infinitesimal flexes of complete bipartite frameworks. Our work generalizes a specific infinitesimal flex to include joined graphs, a family of graphs that contain the complete…
We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we…
In [14], the second named author constructed the bracket invariant [.] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following…