How many crossing changes or Delta-moves does it take to get to a homotopy trivial link?

TL;DR

This paper improves bounds on the number of crossing changes or Delta moves needed to make a link homotopically trivial, using link classification and graph theory, and determines these numbers for 4-component links.

Contribution

It provides a quadratic upper bound on the homotopy trivializing number and a cubic bound on the difference involving triple linking numbers, advancing understanding of link homotopy invariants.

Findings

Quadratic upper bound on the homotopy trivializing number.

Determined the trivializing number for all 4-component links.

Established a cubic bound on the difference involving triple linking numbers.

Abstract

The homotopy trivializing number, \(n_h(L)\), and the Delta homotopy trivializing number, \(n_\Delta(L)\), are invariants of the link homotopy class of \(L\) which count how many crossing changes or Delta moves are needed to reduce that link to a homotopy trivial link. In 2022, Davis, Orson, and Park proved that the homotopy trivializing number of \(L\) is bounded above by the sum of the absolute values of the pairwise linking numbers and some quantity \(C_n\) which depends only on \(n\), the number of components. In this paper we improve on this result by using the classification of link homotopy due to Habegger-Lin to give a quadratic upper bound on \(C_n\). We employ ideas from extremal graph theory to demonstrate that this bound is close to sharp, by exhibiting links with vanishing pairwise linking numbers and whose homotopy trivializing numbers grows quadratically. In the process, we determine the homotopy trivializing number of every 4-component link. We also prove a cubic upper bound on the difference between the Delta homotopy trivializing number of \(L\) and the sum of the absolute values of the triple linking numbers of \(L\).