The forbidden number of a knot

TL;DR

This paper introduces the forbidden number, an invariant measuring the minimum forbidden moves needed to unknot a knot, and explores its relation to existing invariants and bounds for virtual knots.

Contribution

It defines the forbidden number invariant and establishes its connections to known invariants, providing bounds for specific classes of virtual knots.

Findings

Forbidden number is a new invariant for knots.

Bounds are established for certain classes of virtual knots.

Relations to existing invariants are analyzed.

Abstract

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the {\it forbidden number}. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.