Hard diagrams of split links

2024-12-04

arXiv.org

AI Summary

This paper addresses the problem of determining a lower bound on the number of *additional crossings* required to transform a split link diagram into an explicitly split diagram via Reidemeister moves. Prior work established only an Ω(log c) lower bound for diagrams with c crossings. We construct, for arbitrarily large c, a family of split link diagrams with c crossings that necessitate Ω(√c) additional crossings—thereby improving the lower bound to a super-logarithmic, specifically square-root, scale. Methodologically, we integrate bubble tangle theory with the Chambers–Liokumovitch homotopic deformation technique, establishing a novel analytical framework bridging combinatorial topology and Riemannian geometry. Our result provides the first asymptotically tight Ω(√c) lower bound on the unavoidable transient crossing redundancy incurred during splitting, rigorously characterizing the asymptotic growth rate of such redundancy. This significantly widens the gap between previously known upper and lower bounds on link deformation complexity.

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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 $Omega(sqrt{c})$ 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.

Problem

• Study turning split link diagrams into split diagrams
• Show split links need arbitrarily large added crossings
• Prove Ω(√c) extra crossings required for certain diagrams

Innovation

• Uses Reidemeister moves for combinatorial link deformations
• Introduces bubble tangles framework for split links
• Applies Riemannian geometry to convert homotopies into isotopies