Counting graphs induced by Gauss diagrams and families of mutant alternating knots

The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these graphs of small sizes, and we published the number of these graphs as sequence A343358 in the OEIS. The aim of this article is to clarify several subtle theoretical points concerning A343358. Most importantly, we explain why our numbers, produced using graph-theoretical constructions, reflect the number of so-called mutant knots. • We explain in detail what kind of examples is presented by the numbers in our sequence A343358 in the OEIS. • We carefully clarify why the numbers in A343358 differ from those in A002864 and A264759. • We highlight how examples in other datasets correspond to the numbers in A343358.