Related papers: Core groups
The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral…
For a knot diagram $K$, the classical knot group $\pi_1(K)$ is a free group modulo relations determined by Wirtinger-type relations on the classical crossings. The classical knot group is invariant under the Reidemeister moves. In this…
Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that…
We develop a method to show the fundamental group of the double branched covering of a link is not left-orderable by introducing the notion of the coarse presentation. As in the usual group presentations, a coarse presentation is given by a…
A core is said to be a group of central and densely connected nodes which governs the overall behavior of a network. Profiling this meso--scale structure currently relies on a limited number of methods which are often complex, and have…
Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links $\ell$ in a thickened surface $S \times [0,1]$. Their…
Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots…
Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle…
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs…
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a…
We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the…
The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected…
Graph theory is a branch of mathematics in which pair-wise relations between objects are studied. My PhD thesis, supervised by David R. Wood, introduces and investigates a new family of graphs, called link graphs, that generalises the…
We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will…
Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs and graphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We…
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image…
A group invariant for links in thickened closed orientable surfaces is studied. Associated polynomial invariants are defined. The group detects nontriviality of a virtual link and determines its virtual genus.