The full Tutte polynomial from chip-firing data

Definitions and mathematical statement

Matt Baker asks for a chip-firing interpretation of the full Tutte polynomial, beyond Merino's $T_G(1,y)$ theorem [25]. The final extracted sentence, "We made to use some additional geometric structure", is ungrammatical and is not silently replaced by verified text. The aid answers the broad question using auxiliary embedding/root choices, while proving an obstruction to one particular choice-free statistic.

For a finite connected loopless graph with sink $q$, let $\operatorname{Rec}(G,q)$ be its recurrent stable sandpiles and use the Merino level convention \[ \operatorname{lev}(c)=\sum_{v\neq q}c(v)-|E(G)|+\deg(q). \] The Tutte polynomial may be defined by \[ T_G(x,y)=\sum_{A\subseteq E(G)}(x-1)^{r(E)-r(A)}(y-1)^{|A|-r(A)}, \quad r(A)=|V(G)|-\#\operatorname{components}(V(G),A). \] The aid invokes Bernardi's rooted-embedding activity expansion and a level-preserving tree-to-recurrent-sandpile bijection $\eta$. With transported internal activity $J_\eta(c)$, it reports \[ T_G(x,y)=\sum_{c\in\operatorname{Rec}(G,q)}x^{J_\eta(c)}y^{\operatorname{lev}(c)}. \] The statistic depends on the declared auxiliary choices. For plane graphs, the summary describes the $x$-exponent as the level of the sandpile associated to the complementary dual spanning tree, while $y$ records primal level, under compatible dual choices and loop conventions when necessary. These data are not supplied in full detail.

The canonicity obstruction concerns a cycle $C_n$, $n\geq3$, rooted at a vertex. Its recurrent configurations are the all-one nonsink configuration of level one and the $n-1$ configurations with one zero and all other nonsink entries one, of level zero. The aid proves that no rooted-automorphism-invariant nonnegative-integer statistic $J$ can satisfy \[ T_{C_n}(x,y)=\sum_c x^{J(c)}y^{\operatorname{lev}(c)} \] with one positive monomial per configuration. Rooted reflection exchanges some zero-position configurations, forcing repeated monomials, whereas \[ T_{C_n}(x,y)=x+x^2+\cdots+x^{n-1}+y. \] This excludes that invariant positive-monomial form, not every possible chip-firing interpretation or weighted/cancelling expression.

An orientation of the rooted cycle repairs it explicitly. Number the nonsink vertices $1,\ldots,n-1$ along that orientation; assign $J=j$ to the configuration whose zero is at vertex $j$, and $J=0$ to the all-one configuration. The displayed full polynomial follows.

For a connected simple unicyclic graph with sink on its unique cycle and $b$ attached bridges, the aid reports the extension through the natural bridge-reduction correspondence with cycle recurrent configurations. The level is preserved and the exponent becomes $b+J_{\mathrm{cycle}}$. Accordingly \[ T_G(x,y)=x^b\left(x+x^2+\cdots+x^{n-1}+y\right), \] where $n$ is the cycle length. The zero-position description refers to the reduced cycle configuration, not literal zeros on attached trees. This explicit family and the existing choice-dependent theory answer different aspects of the original broad request.

Short English statement

The full Tutte polynomial already has a chip-firing expansion after auxiliary embedding choices. The aid shows why a particular automorphism-invariant monomial statistic cannot work even on rooted cycles. Choosing a cycle orientation supplies an explicit statistic, which extends over attached bridges.

Sources

• Local UnsolvedMath registry, permanent ID $20000994$, source code catalogue entry, supplied complete source record and research summary.
• AIM workshop Generalizations of chip-firing and the critical group, Problem (29), Matt Baker; supplied source https://aimath.org/pastworkshops/chipfiringproblems.pdf. Canonical local record aim-combinatorics-notes.json, notes[118]. No PDF was retrieved.
• Merino's theorem [25], as cited in the original text; Bernardi's rooted-embedding activity expansion and level-preserving bijection, as cited by the aid. No titles, years or full bibliographic entries are supplied; the original literature field is empty.
• Research attempt 1: partial result; broad problem solved at the recorded run, registry status solved and classification SOLVED-IN-LITERATURE; supplied rigor label "proved", novelty confidence "low". The machine-generated aid is not peer reviewed and requires independent verification before citation.