Quadrisecants give new lower bounds for the ropelength of a knot

Denne, Elizabeth; Diao, Yuanan; Sullivan, John M.

doi:10.2140/gt.2006.10.1

Cited by 31 publications

(37 citation statements)

References 19 publications

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“…Since the completion of this paper a generalized version of Theorem 22 was very recently obtained by Denne et al [7], calculating new lower bounds on the ropelength of knots.…”

Section: Theorem 21mentioning

confidence: 97%

Geometric dilation of closed planar curves: New lower bounds

Ebbers-Baumann

Grüne

Klein

2007

Computational Geometry

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Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.

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“…Since the completion of this paper a generalized version of Theorem 22 was very recently obtained by Denne et al [7], calculating new lower bounds on the ropelength of knots.…”

Section: Theorem 21mentioning

confidence: 97%

Geometric dilation of closed planar curves: New lower bounds

Ebbers-Baumann

Grüne

Klein

2007

Computational Geometry

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“…Our distortion bound for knots uses the notion of essential arcs, introduced in [DDS06] as an extension of ideas of Kuperberg [Kup94]. Note that generically a knot K together with a chord pq forms a θ-graph in space; being essential is a topological feature of this knotted graph, as shown in Figure 2.…”

Section: Definitions and Backgroundmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

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confidence: 99%

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The distortion of a knotted curve

Denne

Sullivan

2008

Proc. Amer. Math. Soc.

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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.

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Geometry and Topology

Grechuk

2021

Landscape of 21st Century Mathematics

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Quadrisecants give new lower bounds for the ropelength of a knot

Abstract: 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.

Cited by 31 publications

References 19 publications

Geometric dilation of closed planar curves: New lower bounds

Geometric dilation of closed planar curves: New lower bounds

The distortion of a knotted curve

Geometry and Topology

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