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Im currently reading Matroids: a geometric introduction by Gordon and McNulty. Chapter 9 talks about Tutte polynomials. My question is this Suppose we have two matroids M1 and M2. Both matroids have the same Rank, number of elements in the Ground set, and the same number of Bases. Is there such a thing as two non-isomorphic matroids to have the same Tutte polynomial in rank 1 or 2?? I've been playing around with matroids in Rank 1, rank 2 and rank 3. So far I've only been able to get the same polynomial in rank 3.

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  • $\begingroup$ You should explain what you mean by "have the same Tutte polynomial in rank 1 or 2". Note that there are examples of nonisomorphic 3-connected matroids with the same rank polynomial. (Look up "rotor".) $\endgroup$ Jul 12, 2017 at 20:18
  • $\begingroup$ As mentioned, the Tutte polynomial of a matroid does not, in general, determine the matroid. On the other hand, Anna de Mier's thesis contains a wealth of results concerning matroids and graphs that are completely determined by their Tutte polynomials. $\endgroup$
    – Aaron Dall
    Aug 4, 2017 at 17:54

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Matroids of ranks $1$ and $2$ have simple descriptions, from which one can check that the Tutte polynomial determines the matroid in these cases.

A matroid of rank $1$ is always $\ell \geq 0$ loops together with $m \geq 1$ parallel elements; the corresponding Tutte polynomial is $T=y^{\ell} (x+y+y^2+\cdots+y^{m-1})$. Clearly, $\ell$ and $m$ can be recovered from $T$.

A matroid of rank $2$ is always $\ell \geq 0$ loops together with $k \geq 2$ parallel classes, of sizes $m_1$, $m_2$, ..., $m_k$. I get that the Tutte polynomial is $$y^{\ell} \left[ x^2 + x \sum (1+y+\cdots+y^{m_i-1}) -2x + \mbox{polynomial in $y$} \right].$$ Again, we can read off $\ell$ and $m_1$, ..., $m_k$. I computed (hopefully correctly) using the recursion in Lemma 2.4 here.

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  • $\begingroup$ Thank you! I was not including the loops in the Tutte polynomial, so I couldn't get a better picture of things. This clear some things up, thank you again. Now I have to find what characteristics two matroids need to have to have the same Tutte polynomial but be non-isomorphic. $\endgroup$ Jul 14, 2017 at 0:58

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