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Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1)cot(π/(2q + 1)) (respectively, 2qcot(π/(2q + 1))). Using these calculations, we provide the bounds c₁ ≤ 2/π and c₂ ≥ 5/3cotπ/5 for the constants c₁ and c₂ that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c₁ C(K) ≤ R(K) ≤ c₂ C(K).

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants $c_1,c_2>0$ such that the ribbonlength is bounded above by $c_{1}Cr{(K)}^2$, and also by $c_{2}Cr{(K)}^{3/2}$. We use a different method for each bound. The constant $c_1$ is quite small in comparison to $c_2$, and the first bound is lower than the second for knots and links with $Cr(K)\le$ 12,748.

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent strands $s$ full twists. In this paper, we show that the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(max\{p,q,r\}+|s|r)$ where $p$ and $q$ are positive. Furthermore, if $r\le p-q$, then the ribbonlength of $T_{p,q;r,s}$ is less then or equal to $2(p+(\left|s\right|-1)r)$.

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for $3$-stick unknots with ribbon linking numbers $\pm 1$ and $\pm 3$, and we prove that the minimum folded ribbonlength for $n$-gons with obtuse interior angles is achieved when the $n$-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number $\pm n$ is bounded from above by $2n$.