Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.04403

Profiles of Critical Flat Ribbon Knots

José Ayala
View original source ↗Read source document ↗Export metadata ↓

Abstract

The main open problem in geometric knot theory is to provide a tabulation of knots based on an energy criterion, with the goal of presenting this tabulation in terms of global energy minimisers within isotopy classes, often referred to as ideal knots. Recently, the first examples of minimal length diagrams and their corresponding length values have been determined by Ayala, Kirszenblat, and Rubinstein. This article is motivated by the scarcity of examples despite several decades of intense research. Here, we compute the minimal ribbonlength for some well-known knot diagrams, including the Salomon knot and the Turk's head knot. We also determine the minimal ribbonlength for the granny knot and square knot using a direct method. We conclude by providing the ribbonlength for infinite classes of critical ribbon knots, along with conjectures aimed at relating ribbonlength to knot invariants in pursuit of a metric classification of knots.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

José Ayala. 2025-06-04. Profiles of Critical Flat Ribbon Knots. https://arxiv.org/abs/2506.04403

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗