# Gaps between consecutive untwisting numbers

## TL;DR

This paper investigates the differences between consecutive untwisting numbers of knots, demonstrating that these gaps can be arbitrarily large for certain classes of knots and parameters, revealing new complexity in knot untwisting measures.

## Contribution

It establishes that for any p ≥ 2, the difference between consecutive untwisting numbers can be arbitrarily large, and shows large gaps specifically for torus knots between tu_1 and tu_2.

## Key findings

- Differences between tu_{p-1} and tu_p can be arbitrarily large for p ≥ 2.
- Torus knots exhibit arbitrarily large gaps between tu_1 and tu_2.
- The study reveals significant variability in untwisting numbers across different knots and parameters.

## Abstract

For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$ called the $p$th untwisting number which counts the number of null-homologous twists on at most $2p$ strands required to convert the knot to the unknot. We show that for any $p\geq 2$ the difference between the consecutive untwisting numbers $tu_{p-1}$ and $tu_p$ can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between $tu_1$ and $tu_2$.