A relation between the crossing number and the height of a knotoid

TL;DR

This paper establishes a fundamental inequality relating the crossing number and height of knotoids, providing new bounds and applications for classical knot diagrams and their properties.

Contribution

It proves that the crossing number of a knotoid is at least twice its height and derives bounds for knot bridge lengths using this relation.

Findings

Crossing number ≥ 2 × height for knotoids

Derived lower bounds for crossing number using polynomial invariants

Established an upper bound for bridge length in minimal knot diagrams

Abstract

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.