Linear Upper Bounds on the Ribbonlength of Knots and Links

Abstract: A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length required to tie the given knot type as a folded ribbon knot. The ribbonlength has been conjectured to grow linearly or sub-linearly with respect to a minimal crossing number. Several knot types provide evidence that this conjecture is true, but there is no proof for general cases. In this paper, we show that for any knot or link, the ribbonlength is bounded by a linear function of the crossing number. In more detail, $$ \text{Rib}(K) \leq 2.5 c(K)+1. $$ for a knot or link $K$. Our approach involves binary grid diagrams and bisected vertex leveling techniques.