Folded Ribbon Knot

• Folded Ribbon Knot is a planar model constructed from a thin rectangular strip with a polygonal centerline and prescribed over/under crossing data.
• It introduces the ribbonlength invariant—the ratio of centerline length to ribbon width—which drives optimization in flat folded constructions.
• The theory connects geometric knot theory, framed knot invariants, and paper-band models, offering explicit constructions for unknots, torus, twist, and pretzel knots.

A folded ribbon knot is Kauffman’s planar model of a knot made from a long, thin rectangular strip folded flat in the plane so that its centerline is a polygonal knot diagram with consistent over/under crossing data. Its basic quantitative invariant is the ribbonlength, the scale-invariant ratio of centerline length to ribbon width. The resulting minimization problem asks, for a fixed knot or link type, how efficiently the type can be realized by a flat folded ribbon. In current work, the subject connects geometric knot theory, framed-knot invariants, polygonal projection theory, and paper-band constructions such as Möbius bands and annuli (Denne et al., 2016).

1. Planar model and local geometry

In the standard formulation, one begins with an oriented polygonal knot diagram $K\subset \mathbb{R}^2$, viewed as the image of a piecewise-linear immersion of $S^1$ with prescribed over/under crossing data. If the vertices are $v_1,\dots,v_n$ and the edges are $e_i=[v_i,v_{i+1}]$, then at each vertex $v_i$ the incoming and outgoing edges determine an interior angle $\theta_i\in[0,\pi]$. For a ribbon of width $w>0$, the folded ribbon is constructed by inserting at each vertex with $\theta_i<\pi$ a straight fold line of length

$\frac{w}{\cos(\theta_i/2)}$

centered at $v_i$ and perpendicular to the bisector of $S^1$0, then joining corresponding endpoints of consecutive fold lines by segments parallel to the edges of the knot diagram and at distance $S^1$1 from the centerline. At each fold one must specify whether the outgoing segment passes over or under the incoming segment; this choice is the folding information $S^1$2 (Denne et al., 2016).

An admissible folded ribbon is required to be an immersion except at intended overlaps, with crossing information compatible both with the original knot diagram and with the local overfold/underfold choices. Equivalent formulations describe the object as a flat embedding or immersion of a constant-width strip in the plane with finitely many straight fold lines whose core is a piecewise-linear closed curve representing the knot or link type. In this sense, a folded ribbon knot is simultaneously a planar combinatorial object and a physical paper-band model (Kim et al., 2022).

The local geometry of folds is central. Several later constructions exploit the fact that right-angle folds are especially efficient, and explicit optimization arguments for unknots and twist regions are built from that observation. This suggests that much of the global ribbonlength problem is governed by a constrained packing problem for locally optimal folds rather than by the crossing structure alone (Denne et al., 2022).

2. Ribbonlength, equivalence, and projection complexity

For a folded ribbon realization $S^1$3, the ribbonlength is

$S^1$4

Because both numerator and denominator scale linearly, the quantity is invariant under uniform rescaling, and one often fixes $S^1$5. For a knot or link type $S^1$6, one then defines

$S^1$7

or equivalently the infimum of the core length among all width-one realizations. The ribbonlength problem asks for this infimum as a function of knot or link type (Kim et al., 2022).

A distinctive feature of the theory is that the same underlying knot diagram can support inequivalent folded ribbons with different ribbonlengths. The survey literature distinguishes three notions of folded-ribbon equivalence: knot-diagram equivalence, which fixes only the planar diagram; topological equivalence, which also fixes whether the ribbon is an annulus or a Möbius band; and link equivalence, which further fixes the ribbon linking number. The minimization problem can therefore be posed at several levels of refinement, and the answer may depend on which equivalence relation is imposed (Denne, 2018).

This dependence is visible even in very small examples. For the 3-stick unknot, changing a single fold type changes both the ribbon linking number and the minimal ribbonlength. More generally, the choice of fold pattern may alter the geometry of the boundary polygon and create shorter or longer realizations at fixed diagrammatic complexity. Accordingly, folded ribbon knot theory is not reducible to the planar stick structure of the core alone (Denne et al., 2016).

A related invariant is the projection stick number $S^1$8, the minimum number of straight edges in any planar projection of some space-polygonal representative of $S^1$9. It is natural to ask whether a projection realizing $v_1,\dots,v_n$0 also realizes minimal ribbonlength. A counterexample is provided by two diagrams of the $v_1,\dots,v_n$1 torus knot: one has $v_1,\dots,v_n$2, another uses $v_1,\dots,v_n$3 edges, yet the $v_1,\dots,v_n$4-edge diagram admits a strictly smaller ribbonlength. Passing between the two requires Reidemeister I moves, which change the ribbon linking number by introducing or removing full twists, so the shorter ribbon is not link equivalent to the projection-stick representative. The precise relationship between $v_1,\dots,v_n$5 and minimal link-equivalent ribbonlength therefore remains open (Denne et al., 2016).

3. Extremal results for unknots

The unknot is the setting in which the sharpest exact results are presently known. For an $v_1,\dots,v_n$6-stick unknot represented by a regular $v_1,\dots,v_n$7-gon, a symmetric shrinking argument yields the upper bound

$v_1,\dots,v_n$8

In the construction, one sets $v_1,\dots,v_n$9, shrinks the diagram until adjacent fold segments meet in the center, and computes the side length of the polygonal core from the half-width condition. Because any $e_i=[v_i,v_{i+1}]$0-stick unknot diagram can be deformed to a regular $e_i=[v_i,v_{i+1}]$1-gon under link equivalence, this gives a general bound for every $e_i=[v_i,v_{i+1}]$2-stick unknot (Denne et al., 2016).

The 3-stick case exhibits the strongest dependence on folding information. Up to symmetry, there are two folding patterns for a triangular unknot: either all three folds are the same, or one fold is opposite to the other two. When all three folds are of the same type, the minimum ribbonlength is

$e_i=[v_i,v_{i+1}]$3

and the minimizer is an equilateral triangle. The proof uses the incenter and inradius, together with the classical fact that among triangles of fixed perimeter, the equilateral triangle maximizes the inradius (Denne et al., 2016).

A later refinement reorganizes the 3-stick analysis by ribbon linking number. If the 3-stick unknot has ribbon linking number $e_i=[v_i,v_{i+1}]$4, then the minimum ribbonlength is $e_i=[v_i,v_{i+1}]$5, attained by the equilateral triangle. If the ribbon linking number is $e_i=[v_i,v_{i+1}]$6, then the minimum is $e_i=[v_i,v_{i+1}]$7, again attained in the equilateral case. Thus the same core combinatorics support different sharp minima in different link classes (Denne et al., 2022).

The same paper proves a broader regularity theorem: among convex $e_i=[v_i,v_{i+1}]$8-gons of fixed perimeter whose interior angles all lie in $e_i=[v_i,v_{i+1}]$9, the minimum ribbonlength is achieved exactly by the regular $v_i$0-gon, with value

$v_i$1

It also shows that any annular folded ribbon unknot with ribbon linking number $v_i$2 has minimum ribbonlength $v_i$3. The construction uses $v_i$4 right-angle folds of the same sign, while the lower bound follows from fold counting. These results make the unknot a model case in which ribbonlength can be determined exactly within specified framing data (Denne et al., 2022).

4. Crossing number and asymptotic bounds

A major theme in the subject is the relation between ribbonlength and crossing number. Early general bounds came from grid-diagram methods. For an arbitrary knot $v_i$5, using the grid index estimate $v_i$6, Tian obtained the quadratic upper bound

$v_i$7

The same paper also gave linear upper bounds for torus knots and twist knots and a sharper multiple-folding construction for twist knots (Tian, 2018).

Subsequent work improved the general theory in two directions. One approach, via arc-presentations, gave explicit quadratic bounds with small constants depending on the parity of the arc index; another, via lattice embeddings of regular projections, produced a general $v_i$8 bound. In the formulation stated there, every knot or link satisfies one of two explicit $v_i$9-power estimates, depending on whether a minimal crossing projection contains a Hamiltonian cycle. This established that the general upper exponent could be pushed below $\theta_i\in[0,\pi]$0, although not yet to linear growth (Denne, 2020).

The decisive step came with the use of binary grid diagrams and bisected vertex leveling. Kim, No, and Yoo proved that for any knot or link,

$\theta_i\in[0,\pi]$1

Their construction converts a minimal-crossing diagram into a binary grid diagram, decomposes the grid into six block types, replaces the active blocks by folded “paper-plane” modules of core length $\theta_i\in[0,\pi]$2, and then counts active blocks to obtain the factor $\theta_i\in[0,\pi]$3. In the terminology of that paper, this verifies the upper-bound exponent $\theta_i\in[0,\pi]$4 in the ribbonlength–crossing-number problem for all knots and links (Kim et al., 2024).

The lower-bound side has developed in the opposite direction. Constructions for infinite families with uniformly bounded ribbonlength show that no universal lower bound of the form $\theta_i\in[0,\pi]$5 can hold with $\theta_i\in[0,\pi]$6. In that sense, the lower-bound exponent is forced to be $\theta_i\in[0,\pi]$7. This conclusion is now supported by bounded-ribbonlength families built from narrow-twist or accordion-type constructions (Denne et al., 22 Sep 2025).

5. Explicit constructions for knot and link families

A large part of the literature consists of family-specific constructions that improve general bounds, often by exploiting special tangle or twist structure. For 2-bridge knots and links, an explicit rational-tangle construction realizes each integer tangle $\theta_i\in[0,\pi]$8 by two repeatedly folded ribbons with four free end segments, then connects these blocks according to Conway notation and finally performs the denominator closure. If $\theta_i\in[0,\pi]$9, this yields

$w>0$0

for every 2-bridge knot or link (Kim et al., 2022).

For torus, twist, and pretzel families, progressively sharper bounds have been obtained. The following table records representative proven upper bounds that arise from explicit constructions.

Family	Proven upper bound
General knot or link $w>0$1	$w>0$2
2-bridge knot or link $w>0$3	$w>0$4
$w>0$5-torus link	$w>0$6
Twist knot $w>0$7	$w>0$8
3-strand pretzel link $w>0$9	$\theta_i<\pi$0
$\theta_i<\pi$1-torus knot	$\theta_i<\pi$2
Twist knot	$\theta_i<\pi$3

The first two lines are the universal linear theorem and the 2-bridge theorem already noted. The bounds $\theta_i<\pi$4, $\theta_i<\pi$5, and $\theta_i<\pi$6 come from the 2025 “wrap method,” which localizes large numbers of half-twists in a square region and then closes the diagram with short joins. In particular, applying the twist-knot formula to the figure-eight knot $\theta_i<\pi$7 gives a folded ribbon realization with ribbonlength $\theta_i<\pi$8, which is conjectured there to be the infimum. The uniform bounds $\theta_i<\pi$9 for $\frac{w}{\cos(\theta_i/2)}$0-torus knots and $\frac{w}{\cos(\theta_i/2)}$1 for twist knots come from the “escape-accordion” and half-wrap construction, which compresses arbitrarily many half-twists into bounded ribbonlength (Chen et al., 17 Oct 2025).

Other family results fill in the intermediate development. Grid-diagram constructions produced $\frac{w}{\cos(\theta_i/2)}$2 for torus knots and $\frac{w}{\cos(\theta_i/2)}$3 for twist knots, with a sharper asymptotic twist-knot estimate

$\frac{w}{\cos(\theta_i/2)}$4

for large $\frac{w}{\cos(\theta_i/2)}$5 (Tian, 2018). A later family paper gave $\frac{w}{\cos(\theta_i/2)}$6, $\frac{w}{\cos(\theta_i/2)}$7, $\frac{w}{\cos(\theta_i/2)}$8, and a new construction for general $\frac{w}{\cos(\theta_i/2)}$9-torus knots with $v_i$0 showing

$v_i$1

Because $v_i$2 when $v_i$3, that same paper exhibits torus-knot families with sub-linear $v_i$4 upper bounds (Denne et al., 2020).

Pretzel links have become a particularly important test case. One 2025 construction gives

$v_i$5

depending on the parity pattern of $v_i$6, and more generally

$v_i$7

for any $v_i$8-strand pretzel link. This provides an infinite link family with a uniform bound on infimal folded ribbonlength and also underlies recent tabulations of the best known upper bounds for all prime knots with at most $v_i$9 crossings (Denne, 14 Dec 2025).

6. Framing, band topology, and unresolved problems

A folded ribbon knot is naturally a framed knot. Its ribbon linking number $S^1$00 is the linking number of the central curve with one boundary push-off of the ribbon. This invariant records information that is invisible in the underlying knot diagram alone and is one of the main reasons why different fold patterns on the same core can lead to inequivalent folded ribbons (Denne et al., 2016).

For annular ribbons, the Calugăreanu–White relation takes the form

$S^1$01

while for Möbius ribbons the corresponding formula is

$S^1$02

These formulas place folded ribbon knots in the larger framework of framed-curve geometry and explain why Reidemeister I moves, full twists, and fold choices affect the link class of the ribbon even when the core knot type is unchanged (Denne et al., 2022).

The same framework extends beyond knotted cores to paper Möbius bands and annuli with many half-twists. Hennessey constructed $S^1$03-half-twist paper Möbius bands and annuli with aspect ratio less than $S^1$04, independent of $S^1$05. Denne and Patterson later proved that any multi-twist paper Möbius band can be constructed with aspect ratio $S^1$06 for any $S^1$07, and they used related ideas to obtain bounded ribbonlength families of torus knots and twist knots. These results show that large amounts of twisting can be compressed into bounded planar length-to-width ratio (Hennessey, 2024).

Several central questions remain open. The survey literature asks whether Kauffman’s tight pentagonal trefoil model minimizes ribbonlength, whether a ribbonlength minimizer must ever use strictly more sticks than the projection-stick index, and how minimal ribbonlength behaves within fixed topological or link-equivalence classes. More recent work asks whether the constant $S^1$08 in the universal bound $S^1$09 can be improved, and specific constructions conjecture exact values such as $S^1$10 for the figure-eight knot. Exact infima are therefore known in only a restricted range of cases, while the interaction among folding information, ribbon linking number, and combinatorial diagram type continues to drive the subject’s development (Denne, 2018).