Hard diagrams of split links

TL;DR

This paper demonstrates that transforming certain split link diagrams into split diagrams can necessitate a large number of additional crossings, establishing lower bounds on the complexity of such transformations.

Contribution

It introduces a family of split link diagrams requiring arbitrarily many extra crossings to achieve splitting via Reidemeister moves, using bubble tangles and geometric techniques.

Findings

Existence of split link diagrams with arbitrarily large minimal crossing additions

Lower bounds of order (\u221a{c}) on crossing numbers needed for splitting

Application of bubble tangles and geometric methods to knot theory 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 $\Omega(\sqrt{c})$ 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.