The paper provides bounds for the ropelength of a link in terms of the crossing numbers of its split components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial improvement on previous results. The proof depends essentially on writing links in terms of their arc-presentations, and has as a key ingredient Bae and Park's theorem that an n-crossing link has an arc-presentation with less than or equal to n+2 arcs.
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…
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…
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 …
We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a few knots. We also prove that we can always find at least one tangle from the set of four tangles, in any triple-crossing projections of any non-trivial knot or non-split link. In the last section, we enumerate and generate tables of minimal delta-diagrams for all prime knots up to the delta-crossing number equal to four. We also give a concise survey about kn…
We develop a word mechanism applied in knot and link diagrams for the illustration of a diagrammatic property. We also give a necessary condition for determining incompressible and pairwise incompressible surfaces, that are embedded in knot or link complements. Finally, we give a finiteness theorem and an upper bound on the Euler characteristic of such surfaces.
Bae and Park found an upper bound on the arc index of prime links in terms of the minimal crossing number. In this paper, we extend the definition of the arc presentation to spatial graphs and find an upper bound on the arc index $α(G)$ of any spatial graph $G$ as $$α(G) \leq c(G)+e+b,$$ where $c(G)$ is the minimal crossing number of $G$, $e$ is the number of edges, and $b$ is the number of bouquet cut-components. This upper bound is lowest possible.
We answer a question posed by Fielder in [1] concerning two notions of crossing number for algebraic knots $K$ under Hopf fibration, one topological, denoted $h(K)$, the other coming from the realization of such knots around complex singularities, denoted $C_{alg}(K)$. We show that $C_{alg}(K)-h(K)$ can be arbitrarily large. We also give an upper bound for $h$ of some families of knots such as torus knots $T(2,n)$, twist knots and their mirror images.
Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by $(m - 2)^{2} - 2$ if $m$ is odd, and $(m - 2)^{2} - (m - 3)$ if $m$ is even.
This paper gives new upper bounds on the stick numbers of the knots $9_{18}$, $10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$, $10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\leq 5$, completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.
In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $α(L)$ for any knot or non-split link $L$, which are $b(L) \leq \frac{1}{2} c(L) + 1$ and $α(L) \leq c(L) + 2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.
An $n$-crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k \in \mathbb{N}$ and all links $L$, the inequality $$c_{2k+1}(L) \geq \frac{2g(L) + r(L)-1}{k^2}$$ holds, where $c_{2k+1}(L)$, $g(L)$, and $r(L)$ are the $(2k+1)$-crossing number, $3$-genus, and number of components of $L$ respectively. This result is used to prove a new bound on the odd crossing numbers of torus knots and generalizes a result of Jablonowski. We also prove a new upper bound on the $5$-crossing …
Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knot…
Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings
Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa's table of knotted sur…
Previous work used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. The present work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygona…
Our goal is to systematically compute the $\mathbb{C}P^2$-genus of all prime knots up to 8-crossings. We obtain upper bounds on the $\mathbb{C}P^2$-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to $\mathbb{C}P^2$. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the $\mathbb{C}P^2$-genus of all but 2 of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the $\mathbb…
INRIA Université Côte d’Azur（法国蔚蓝海岸大学国家信息与自动化研究所）；LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris（法国巴黎高等信息与机械学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学院）
The unknotting number, hard unknot diagrams, and reinforcement learning
Google DeepMind, London, UK（谷歌深思（DeepMind），伦敦，英国）；Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK（牛津大学数学研究所）
