Completeness conjecture for the Bollobás–Riordan polynomial of lens-space Heegaard graphs

For a lens space $L(p,q)$, let $G_{p,q}$ denote its Heegaard graph, and let $R_{G_{p,q}}$ be the Bollobás–Riordan polynomial of that graph. Two lens spaces are homeomorphic exactly when their Heegaard graphs have the corresponding homeomorphism-class parameters.

Bollobás–Riordan completeness conjecture. The Bollobás–Riordan polynomial of the Heegaard graph of lens spaces is a complete invariant for lens spaces.

The Bollobás–Riordan polynomial specializes to the Tutte polynomial and therefore records richer embedding information. Computed exponent patterns distinguish the orbits $\{\pm q^{\pm1}\}$, but completeness for all lens spaces remains open.