Counterexamples to the Two-Bases Conjecture for the Tutte Polynomial
We disprove the two-bases conjecture of Chávez-Lomelí, Merino, Noble, and Ramírez-Ibañez (Conjecture 5.8 in *Some Inequalities for the Tutte Polynomial*), which asserts that \(\max\{T_M(2,0),T_M(0,2)\}\ge T_M(1,1)\) whenever a matroid contains two disjoint bases or its ground set is the union of two bases. We construct a connected, self-dual, rationally representable matroid \(M\) of rank \(32\) on \(64\) elements whose ground set is partitioned into two bases, but for which \[ T_M(2,0)=T_M(0,2)=263656235940864 <275621102802945=T_M(1,1). \] The construction extends to an explicit infinite…
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