Segment number of knots

Abstract

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number.

Formerly titled "Edge number of knots and links". 8 pages, 5 figures. To appear in Journal of Knot Theory and Its Ramifications