Elizabeth Denne, Yuanan Diao, John M Sullivan
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.
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It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we prove that every (nontrivial tame) knot has an alternating quadrisecant. This result had applications to the total curvature, second hull and ropelength of knots.
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.
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the quadrisecant approximation of the original knot. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self intersections while the quadrisecant approximation of the other knot is a knot with different knot t…
It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.
On the Minimum Ropelength of Knots and Links
该研究证明了任意纽结或链环类型中绳长极小值子的存在性与正则性，改进了非平凡纽结绳长下界，还为小型纽结与链环建立了含紧界的新绳长界。
We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in R^3 which is a knotted torus must have degree at least eight.
Let $K$ be a polygonal knot in general position with vertex set $V$. A \emph{generic quadrisecant} of $K$ is a line that is disjoint from the set $V$ and intersects $K$ in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot $K$ in general position. This upper bound is in terms of the number of edges of $K$.
The ropelength of a knot is the quotient of its length by its thickness. We consider a family of energy functions for knots, depending on a power p, which approach ropelength as p increases. We describe a numerically computed trefoil knot which seems to be a local minimum for ropelength; there are nearby critical points for the p-energies, which are evidently local minima for large enough p.
Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.
Symmetric Criticality for Tight Knots
本研究证明绳长临界纽结的对称临界性定理，指出具对称表示的纽结/链环存在同对称性的绳长临界构型，并据此构造了异于对应类型绳长极小构型的新临界实例。
For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number.
A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the ropelength of the link from below. Thus in the case an alternating link has a maximum braid index proportional to its minimum crossing number, such as the $T(2,2n)$ torus link, then the ropelength of the link is bounded below by a constant multiple of its minimum crossing number. However if the maximum braid index of a link is small compared to its crossing …
The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope needed to tie a given knot. Theoretical upper and lower bounds have been established for the asymptotic relationship between crossing number and ropelength and stronger bounds have been conjectured, but numerical bounds have only been calculated exhaustively for knots and links with up to 11 crossings, which are not sufficiently complex to test these conjectures. Existing ropelength measurements also ha…
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which we can see small trefoil and figure-eight summands and use Monte Carlo simulations to approximate the relative probabilities of these structures as a function of the number of edges.
Ropelength and writhe quantization of 12-crossing knots
本文测量了2176个12交叉结的绳长，发现紧致交错结和非交错结的空间缠绕数分别呈现接近4/7和4/3倍数的量子化现象，并揭示了绳长与双曲体积、空间缠绕数与Rasmussen s不变量的正相关性。
Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot $K$ in terms of its crossing number $c(K)$ as follows: $\mbox{Len}(K) \leq \min \left\{ \frac{3}{4}c(K)^2 + 5c(K) + \frac{17}{4}, \, \frac{5}{8}c(K)^2 + \frac{15}{2}c(K) + \frac{71}{8} \right\}.$ The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the m…
Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot $K$ indicates the minimum length necessary to construct $K$ in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of knot energies measuring the complexity of knot conformation. The minimum ropelength $\mbox{Rop}(K)$ is the minimum len…
The ropelength conjecture of alternating knots
本文证明了长期存在的交错纽结绳长猜想，即存在常数$b_0>0$，使得任何交错纽结$K$的绳长$R(K)$至少与其交叉数$Cr(K)$成正比（$R(K)\ngeq b_0Cr(K)$）。
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave bett…
Ropelength-minimizing concentric helices and non-alternating torus knots
本文研究非交替环面纽结的同心螺旋结构，通过优化几何与组合特性，为3到39个螺旋系统找到高效构型，并为更大系统提出优化构造，将非交替环面纽结的ropelength上下界比率从29降至1.4-3.8。
Knot Tightening By Constrained Gradient Descent
该研究提出约束梯度下降法求解固定厚度下近似长度最小的打结多边形，推导了相关理论准则，得到379个覆盖指定交叉数范围的素纽结与链环的绳长结果及自接触图谱，提升了纽结绳长精度并首次发表链环绳长数据。
Ropelength-Filtered Knot Spaces
研究通过绳长过滤空间分析结类型，证明绳长是变形持久性的首次出现层级，定义了理想层和相关不变量，探讨了纯合并Vietoris-Rips滤泡的结构。
Ropelength-Filtered Knot Densities
研究无约束和绳长-窗口化的p-密度，揭示其退化现象，并通过约束条件改进理论，探讨其单调性和存在性。
Density and Compression on Lattice-Knot Merge Trees
本文建立了格点纽结的离散p-密度和压缩半径轮廓框架，通过长度过滤和局部移动系统研究其性质，并作为案例分析了八字结的密度与压缩半径变化。
Geometric densities and compression radii of knot types
研究嵌入三维欧几里得空间中的闭合曲线的尺度不变几何量，关注其在固定结型下的优化行为。定义D-密度、D-压缩半径及打包比，探讨其在优化后的分离特性。
Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links
本文研究T(Q,Q) torus链接的绳长界，推导出更严格的下界，并展示通过几何优化可使绳长接近下界，为特定结类的绳长界研究提供了重要进展。
Ropelength-Filtered Essential Surface Spaces
该研究为纽结外部本质曲面建立几何有限性理论，通过构造有界规范分层码等方法，证明有界几何切片的同痕类有限，拓扑可由有限几何数据恢复，还得出扭结窗口等结果且框架无需三角剖分，补充了多种理论方法。
We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $K_i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation. These results contrast with the situatio…
We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the space of knots as a subspace of what we call the n-th mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computati…
Milnor invariants and thickness of spherical links
该研究将经典纽结绳长与不变量的关联拓展至高维球面链，利用Massey乘积分析Milnor不变量，得到其关于厚度的最优渐近界并揭示多项式-指数二分性，解决了Freedman和Krushkal的相关问题。
We define an infinite sequence of new invariants, delta_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the delta_n give obstructions to a 3-manifold fibering over S^1 and to a 3-manifold being Seifert fibered. Moreover, we s…
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