The conjecture that knot-generated graphs differ from graphs with polynomial one

Let ${\cal C}_1$ be the set of graphs isotopic to planar graphs, ${\cal C}_2$ the set of graphs isotopic to graphs obtained from planar graphs by the specified moves, ${\cal C}_3$ the set of graphs isotopic to graphs obtained from knots, including the unknot, by those moves, and ${\cal C}_4$ the set of graphs whose polynomial $P$ equals $1$. The known inclusions are ${\cal C}_1\subseteq{\cal C}_2\subseteq{\cal C}_3\subseteq{\cal C}_4$.

The conjecture that ${\cal C}_3$ and ${\cal C}_4$ differ.

$${\cal C}_3\neq{\cal C}_4.$$

This asserts that not every graph with polynomial $P=1$ can be obtained from a knot by the specified moves. The paper presents the question alongside the established inclusions but gives no proof or resolution of this separation.