TL;DR
This paper investigates various measures of how non-alternating a link is, demonstrating that differences between these measures can be arbitrarily large and that some are not directly comparable.
Contribution
It introduces and compares multiple alternating distances, showing their differences can be unbounded and establishing non-comparability of certain measures.
Findings
Differences between certain alternating distances can be arbitrarily large.
The alternation number and the alternating genus are not comparable.
The study provides examples of links with large differences in alternating distances.
Abstract
An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable.
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