Knotted ribbons form an important topic in knot theory. They have applications in
natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be
obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An
important problem in knot theory is to study the minimal ratio of length to width of a
flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been
conjectured that the ribbonlength has an upper bound and a lower bound which are both
linear in the crossing number of the knot.
In the first part of the paper, we use grid diagrams to construct flat knotted ribbons
and prove an explicit quadratic upper bound on the ribbonlength for all non-trivial knots.
We then improve the quadratic upper bound to a linear upper bound for all non-trivial
torus knots and twist knots. Our approach of using grid diagrams to study flat knotted
ribbons is novel and can likely be used to obtain a linear upper bound for more general
families of knots. In the second part of the paper, we obtain a sharper linear upper bound
on the ribbonlength for nontrivial twist knots by constructing a flat knotted ribbon via
folding the ribbon over itself multiple times to shorten the length.