Hard diagrams of split links

TL;DR

This work proves the existence of hard split links with arbitrarily large crossing-complexity by constructing a family $L(n,n{+}1)$ consisting of two linked torus knots and an enclosing unknot. It introduces the crossing-augmentation measure Add and the crossing-complexity $\textbf{CC}(\mathcal{D})$, and shows that any sequence of Reidemeister moves transforming the initial diagram $\mathcal{D}(n,n{+}1)$ into a split diagram must pass through a diagram with at least $2n^2 + \frac{2}{3}n$ crossings, i.e., $\textbf{CC}(\mathcal{D}(n,n{+}1))\ge 2n^2 + \frac{2}{3}n$. The proof blends a sweep-out construction of $2$-spheres from a projected unknot, the bubble-tangle framework, and Chambers–Liokumovich-style homotopy-to-isotopy lifting, adapted to diagrams evolving under Reidemeister moves. These results establish a super-constant lower bound on crossing-complexity for split links and illuminate gaps between known lower and upper bounds in knot diagram transformations.

Abstract

Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister moves, one sometimes needs to insert excess crossings, there are significant gaps between the best known lower and upper bounds on the required number of these added crossings. In this article, we study the problem of turning a diagram of a split link into a split diagram, and we show that there exist split links with diagrams requiring an arbitrarily large number of such additional crossings. More precisely, we provide a family of diagrams of split links, so that any sequence of Reidemeister moves transforming a diagram with $c$ crossings into a split diagram requires going through a diagram with $Ω(\sqrt{c})$ extra crossings. Our proof relies on the framework of bubble tangles, as introduced by the first two authors, and a technique of Chambers and Liokumovitch to turn homotopies into isotopies in the context of Riemannian geometry.

Figures (21)

• Figure 1: The three Reidemeister moves RI, RII, and RIII.
• Figure 2: The Goeritz culprit: using Reidemeister moves in $\mathbb{S}^2$, one must add at least one extra crossing to untangle this unknot diagram.
• Figure 3: The link diagram $\mathcal{D} (p,q)$, $(p,q) = (7,8)$: two linked torus knots $T_{7,8}$ and an unknot $U$.
• Figure 4: Critical times corresponding to Reidemeister moves RII (left) and RIII (right).
• Figure 5: Definition of our homotopies from the sequence of Reidemeister moves $R$.

Theorems & Definitions (24)

• Conjecture 1.1
• Theorem 1.2
• Remark 1.3
• Remark 2.1
• Lemma 2.2
• proof
• Proposition 3.1
• Definition 3.2: Chambers, Liokumovich Chambers_homo-isotopy
• Theorem 3.3: Chambers, Liokumovich Chambers_homo-isotopy
• Remark 3.4