selfcrossings

Selfcrossings refer to instances in a graph or diagram where an edge or line intersects itself, creating a crossing point that is not part of the intended connections between vertices or nodes. This phenomenon is commonly observed in various fields, including graph theory, knot theory, and molecular biology, where the study of molecular structures such as DNA often involves analyzing selfcrossings to understand spatial arrangements.

In graph theory, a selfcrossing occurs when an edge loops back upon itself, creating a crossing that is not explicitly defined by the graph’s adjacency list. Such crossings can complicate the visualization and analysis of graphs, particularly in complex networks where edges may overlap or intertwine. To avoid ambiguity, graphs are often drawn in a planar manner, where edges do not cross unless necessary, though this is not always possible. Non-planar graphs, which inherently contain selfcrossings or edge crossings, require alternative representations, such as using three-dimensional layouts or embedding techniques.

In knot theory, selfcrossings are fundamental to defining and studying knots. A knot is a closed loop of string or rope that can be manipulated in three-dimensional space without cutting or gluing, and its properties are often described by its crossing diagram—a two-dimensional projection where selfcrossings represent the points where the knot intersects itself. The number and arrangement of crossings influence the knot’s classification, with simpler knots having fewer crossings. Techniques like the Reidemeister moves allow mathematicians to manipulate crossing diagrams to determine whether two knots are equivalent.

In molecular biology, selfcrossings appear in the study of nucleic acids, such as DNA, which can form complex three-dimensional structures. When DNA strands loop or intertwine, selfcrossings can occur in their spatial configurations, affecting processes like replication, transcription, and packaging within the cell. Analyzing these crossings helps researchers understand how genetic material is organized and how topological constraints influence biological function. Tools like gel electrophoresis and computational modeling are used to visualize and study these interactions.