Startknotens

Startknotens is a theoretical construct in knot theory and graph theory that refers to a marked knot diagram or knot-graph with a designated starting crossing or node. The designation of a start point allows a canonical traversal of the diagram, producing a sequence of crossings or edges that can be used to encode the knot in a linear form. By fixing a traversal rule, such as following the strand at each crossing according to a consistent over/under rule, one obtains a word or code that represents the knot with reference to the starting point.

Construction: To form a Startknotens, begin with a standard knot diagram or knot-graph. Choose a crossing (or node) to serve as the start. Assign a direction to traverse the diagram, and apply a local rule at each crossing to decide the next step. Continue until the traversal returns to the start. The resulting sequence is the Startknotens code. If different starting points are used, one obtains cyclic shifts of the same underlying structure; invariants can be defined by considering all shifts.

Properties: Startknotens allow comparison by canonical form, selecting the start that minimizes a chosen length or complexity. Some invariants of the underlying knot are retained, while others depend on the chosen start; defining a derived invariant often requires ignoring cyclic shifts or choosing a standard criterion for the start.

Applications: They are used in thought experiments, algorithmic knot recognition, and education to illustrate how a marked starting point affects knot representation. The approach helps in designing deterministic parsing algorithms for knot diagrams.

Etymology: The term combines Start (beginning) and Knoten (knot) in German, reflecting the notion of a knot with a designated starting point.