Forbidden detour number on virtual knot

TL;DR

This paper introduces the forbidden detour number as a new measure for virtual knots, showing it as an unknotting operation and providing bounds based on existing invariants.

Contribution

It defines the forbidden detour number for virtual knots and establishes bounds relating it to crossing number and affine index polynomial coefficients.

Findings

Forbidden detour move is an unknotting operation for virtual knots.

Bounds on the forbidden detour number are given in terms of crossings and polynomial coefficients.

Abstract

We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot diagram. Some upper and lower bounds on the forbidden detour number are given in terms of the minimal number of real crossings or the coefficients of the affine index polynomial of the virtual knot.