Rotors in graphs and rank polynomial (Tutte polynomial)

Question

I am studying the $Rank$ $polynomial$ through matroid theory. I have seen that the rank polynomial doesn't determine the graph. In fact, as the the cycle matroid of a graph can distinguish the graph only if it is $3-$connected, one can choose two different graphs (not $3-$connected) that have the same cycle matroid and therefore the same rank polynomial. Moreover, it can be shown that there are also 3-connected graphs, non isomorphic (also the cycle matroid are not isomorphic because 3-connected graphs are distinguished by the cycle matroid). This can be seen using $rotors$ (if anyone is interested, it is in Godsil-Royle, page 363). This is seen creating two graphs (with the same number of edges and vertex) and a bijective map between the edges that preserves $size$ and $rank$. Unfortunately, I don't understand why such a map is not an isomorphism of a matroid. Can someone tell me what I am missing?

Answer 1

I might be wrong, but I think there is something strange in the idea of the proof that I read in the book of Godsil and Royle. I believe that it is more clear the proof of Tutte written in "Codichromatic graphs, J. Combinatorial Theory Ser. B 16 (1974), 168–174". The Tutte's idea seems a little bit different, but after making some small changes it is satisfying.