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.

Abstract. The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that any closed curve has distortion at least π/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5π/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.Gromov introduced the notion of distortion for curves as the supremal ratio of arclength to chord length. (See [Gro78], [Gro83, p. 114] and [GLP81,.) He showed that any closed curve has distortion δ ≥ π / 2 , with equality only for a circle. He then asked whether every knot type can be built with, say, δ ≤ 100.As Gromov knew, there are infinite families with such a uniform bound. For instance, an open trefoil (a long knot with straight ends) can be built with δ < 10.7, as follows from an explicit computation for a simple shape. Then connect sums of arbitrarily many trefoils-even infinitely many, as in Despite such examples, many people expect a negative answer to Gromov's question. We provide a first step in this direction, namely a lower bound depending on knottedness: we prove that any nontrivial tame knot has δ ≥ 5π / 3 , more than three times the minimum for an unknot.To make further progress on the original question, one should try to bound distortion in terms of some measure of knot complexity. Examples such as Figure 1 show that crossing number and even bridge number are too strong: distortion can stay bounded as they go to infinity. Perhaps it is worth investigating hull number [CKKS03,Izm06].Our bound δ ≥ 5π / 3 arises from considering essential secants of the knot, a notion introduced by Kuperberg [Kup94] and developed further in [DDS06]. There, we used the essential alternating quadrisecants of [Den04] to give a good lower bound for the ropelength [GM99, CKS02] of nontrivial knots.The main tool in [DDS06] was a geometric characterization of a borderlineessential arc (quoted here as Theorem 1.1), showing that its endpoints are part of an essential trisecant. This result captures the intuition that in order for an arc to become essential, it must wrap around some other point of the knot; but it also demonstrates that secants to that other point are themselves essential. This theorem will be important for our distortion bounds as well.

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 [Formula: see text] such that the ribbonlength is bounded above by [Formula: see text], and also by [Formula: see text]. We use a different method for each bound. The constant [Formula: see text] is quite small in comparison to [Formula: see text], and the first bound is lower than the second for knots and links with [Formula: see text] 12,748.

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.

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 R n where each simple closed curve has an odd number of inscribed square-like quadrilaterals.