Search

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 $\pm1$ 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 $2n$., Comment: 36 pages, 21 figures

Let $\Delta$ denote a non-degenerate $k$-simplex in $\mathbb{R}^k$. The set $\text{Sim}(\Delta)$ of simplices in $\mathbb{R}^k$ similar to $\Delta$ is diffeomorphic to $O(k)\times [0,\infty)\times \mathbb{R}^k$, where the factor in $O(k)$ is a matrix called the {\em pose}. Among $(k-1)$-spheres smoothly embedded in $\mathbb{R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\text{Sim}(\Delta)$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in O(k)$. Further, the intersection of $\text{Sim}(\Delta)$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $O(k)$ via the pose map. This gives a high dimensional generalization of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces., Comment: 20 pages, 2 figures. arXiv admin note: text overlap with arXiv:2103.07506 New version has correct term for $k$-simplex and other minor corrections

The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between continuous and $C^1$-smooth Jordan curves. Here, in a generalization of the square-peg problem, we consider embedded curves in space, and ask if they have inscribed quadrilaterals with equal sides and equal diagonals. We call these quadrilaterals "square-like". We give a regularity class (finite total curvature without cusps) in which we can prove that every embedded curve has an inscribed square-like quadrilateral. The key idea is to use local data to show that short enough arcs have small curvature, thus ruling out small squares. This allows us to successfully use a limiting argument on approximating curves., Comment: 11 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.6174

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold $M$ in Euclidean space, we can find a dense set of smooth embeddings of $M$ for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in $\mathbb{R}^n$ where each simple closed curve has an odd number of inscribed square-like quadrilaterals., Comment: 30 pages, 4 figures. arXiv admin note: text overlap with arXiv:1402.6174. Version 2 corrects some minor errors

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 ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\geq q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot Cr(K)^{1/2}$. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number., Comment: 33 pages, 26 figures. Second version corrects minor errors as well as Theorem 4: $(p,q)$ torus links with $p\geq q\geq 2$ have folded ribbonlength bounded above by $2p$

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 $c_1\cdot Cr(K)^2$, and also by $c_2\cdot Cr(K)^{3/2}$. 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 $Cr(K)\leq$ 12,748., Comment: 19 pages, 7 figures. Revision 1: Correction to Theorem 2

In 2000, Thomas Fink and Young Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The ends of a neck tie can be joined together, which gives a physical model of a mathematical knot that we call a tie knot. In this paper we classify the knot type of each of Fink and Mao's 85 tie knots. We describe how the unknot, left and right trefoil, twist knots and $(2,p)$ torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating., Comment: 22 pages, 17 figures, 3 tables. Some minor errors corrected

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 most part, these are upper bounds of ribbonlength of twist knots and certain families of torus knots. We discuss result of G. Tian, which give upper bounds of ribbonlength in terms of crossing number. In addition, it turns out the choice of fold affects the ribbonlength. We end with a discussion of three different types of folded ribbon equivalence and give examples illustrating their relationship to ribbonlength., Comment: 12 pages, 10 figures. arXiv admin note: text overlap with arXiv:1602.08084

A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot., Comment: 17 pages, 7 figures

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 $n\cot(\pi/n)$ for the ribbonlength of $n$-stick unknots. We prove that the minimum ribbonlength for a 3-stick unknot with the same type of fold at each vertex is $3\sqrt{3}$, and such a minimizer is an equilateral triangle. We end the paper with a discussion of projection stick number and ribbonlength., Comment: 16 pages, 16 figures

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily $C^1$-small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of "special inscribed configurations" inside families of spheres embedded in $\mathbb{R}^n$ using differential topology. For instance, there is a $C^1$-dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a $C^1$-dense family of smooth embedded $(n-1)$-spheres in $\mathbb{R}^n$ where each sphere has a family of inscribed regular $n$-simplices with the homology of $O(n)$., Comment: 32 pages, 8 figures. This paper has been subdivided and will be published as three different papers (cited in text). This version is being left on arxiv as a service to the reader; it will not be published

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot., Comment: 12 pages, 3 figures; final version, to appear in "Discrete Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 2008

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil knot. Our argument uses the existence of a shortest essential secant and a characterization of borderline-essential arcs., Comment: 5 pages, 7 figures; substantial improvements: bound is now 5pi/3 and applies to all tame knots

Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp., Comment: v3 is the version published by Geometry & Topology on 25 February 2006