Positive braids minimize ascending number

TL;DR

This paper proves that for knots formed from positive braids, the ascending number equals the unknotting number, and explores whether this property extends to positive knots, supported by data on low-crossing knots.

Contribution

It establishes the equality of ascending and unknotting numbers for positive braid knots and investigates the extent of this property among low-crossing knots.

Findings

Ascending number equals unknotting number for positive braid knots

Data suggests similar property may hold for positive knots

Limited number of low-crossing hyperbolic, alternating knots exhibit this property

Abstract

A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the ascending number of the knot. The ascending number is bounded below by the unknotting number. We show that for knots obtained as the closure of a positive braid, the ascending number equals the unknotting number. We also present data indicating that a similar result may hold for positive knots. We use this data to examine which low-crossing knots have the property that their ascending number is realized in a minimal crossing diagram, showing that there are at most 5 hyperbolic, alternating knots with at most 12 crossings with this property.