Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for (2,q)-torus links, twist knots, and pretzel links. This paper improves upper bounds on ribbonlength for certain alternating links.
problem Finding tighter bounds on the ribbonlength of alternating links.
method Proved a new upper bound for alternating links with bipartite dual graphs.
result Improved upper bounds on ribbonlength for specific links.
This survey reviews 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 ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the mos…
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 ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of ncot(π/n) for the ribbonlength of $…
This paper finds bounds on the ribbonlength of knots and links with up to 9 crossings.
problem Finding the infimal folded ribbonlength of knots and links.
method Analyzing pretzel links and constructing knots to find bounds on ribbonlength.
result Uniform bounds on infimal folded ribbonlength for certain link families.
Study minimum ribbonlength of immersed flat knots and links.
problem Finding the minimum ribbonlength for immersed planar knots and links.
method Embedding into disk diagram space to find length minimizers.
result Computed minimal ribbonlength for some knot and link diagrams.
Upper bounds on ribbonlength of various knots, showing linear and sub-linear behavior.
problem Estimating the ribbonlength of different types of knots.
method Using Kauffman's model of folded ribbon knots, we derive upper bounds on ribbonlength for specific knot types.
result Upper bounds on ribbonlength are linear in crossing number for some knots and sub-linear for others.
Paper calculates the ribbonlength of twisted torus knots.
problem Determining the ribbonlength of twisted torus knots.
method Analyzes the ribbonlength of twisted torus knots Tp,q;r,s and provides upper bounds. result Ribbonlength of Tp,q;r,s is bounded by 2(max{p,q,r}+∣s∣r) and 2(p+(∣s∣−1)r) under specific conditions. This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. Improved linear upper bound for ribbonlength of knots.
problem Estimating the ribbonlength of knots and links.
method Using four-page open book decompositions and spanning trees of checkerboard graphs, constructing a four-page presentation with at most 2c(K) arcs.
result Proved that ribbonlength is bounded above by the four-page index, leading to the linear bound Rib(K) ≤ 2c(K).
Study on ribbonlength and crossing number for folded ribbon knots.
problem Understanding the relationship between ribbonlength and crossing number for folded ribbon knots.
method Used two different methods to establish upper bounds on ribbonlength in terms of crossing number.
result Found constants c1 and c2 such that ribbonlength is bounded above by c1⋅Cr(K)2 and c2⋅Cr(K)3/2, respectively. Proves bounds on ribbonlength for various knot types.
problem Finding bounds on the infimal folded ribbonlength of knot types.
method Applied techniques to multi-twist Möbius bands and knot constructions.
result Established bounds for (2,q) torus knots and twist knots. The paper calculates minimal ribbonlength for various knots.
problem Tabulation of knots based on energy criterion.
method Direct method for computing minimal ribbonlength.
result Computed minimal ribbonlength for specific knots.
Paper proves ribbonlength grows linearly with knot complexity.
problem Proving ribbonlength grows linearly with knot complexity.
method Binary grid diagrams and bisected vertex leveling techniques.
result Ribbonlength is bounded by a linear function of the crossing number.
Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised 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 clo…
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
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 …