The problem of which Gauss diagram can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances" and the sufficient condition is based on Jordan curve Theorem. Further, using matrixes we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.Mathematics Subject Classifications: 57M25, 14H50.
The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial p ∈ N[x], and any directed finite bipartite graph can be considered as a polynomial p ∈ N[x, y], and vise verse. We also show that the multiplication in semirings N[x], N[x, y] correspondences to a operations of the corresponding graphs which looks like a "perturbed" products of graphs. As an application, we give a new point of view to dividing in semirings N[x], N[x, y]. Finally, we endow the set of all bipartite graphs with the Zariski topology.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.