Author

W. Whistler

View on Semantic Scholar

2 papers indexed here

We haven’t gathered this author’s papers yet. Follow them and we’ll fetch their work.

Not the right person? Other researchers publish under this name.

Jul 2026

Counting spanning quasi-trees of ribbon graphs: determinants and #P-completeness

A quasi-tree of a connected ribbon graph is a spanning ribbon subgraph with exactly one boundary component; quasi-trees play the role of spanning trees in the topological graph theory of embedded graphs. We prove that counting them is #P-complete under polynomial-time Turing reductions, already for bouquets. The proof identifies every nonempty framed chord diagram, up to natural identifications, with a 4-regular map equipped with a distinguished A-trail, in such a way that quasi-trees correspond to A-trails, whose counting is #P-complete by a theorem of Ge and \v{S}tefankovi\v{c}. Through the framed Cohn-Lempel equality the count is also an interlace-polynomial evaluation - $q(H;2,1)$, the number of full-rank induced subgraphs of the looped circle graph $H$ of the diagram - placing it on the line $y=1$ left open in the complexity classification of Bl\"aser and Hoffmann; a cloning argument then makes every fixed rational point of that line, other than the trivial $(1,1)$, #P-hard on looped circle graphs, even when a framed chord representation is supplied. On the tractable side, the same GF(2) model yields short proofs of the known determinantal cases: for orientable ribbon graphs the count is a determinant, essentially the Matrix-Quasi-tree Theorem of Merino, Moffatt and Noble, proved here via Bouchet's principal unimodularity, and for bouquets with exactly one non-orientable loop it is a sum of two orientable determinants, equivalent by a rank-one determinant identity to the determinant formula of Deng, Jin and Yan.

W. Whistler · arXiv.org · 0 citations

Preprint Jul 2026

Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank

We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter $f$ with $f(\varnothing)=1$ has exponentially bounded edge-connection rank if and only if it is a mixed partition function; moreover, the model may be chosen with its numbers of even and odd colours explicitly bounded in terms of the rank bound. From $f$ we construct a connection category, a rigid symmetric $\mathbb{C}$-linear monoidal category whose morphism spaces have the connection ranks as dimensions and whose trace pairings are nondegenerate. The rank hypothesis forces moderate tensor growth, and a recent theorem of Etingof and Penneys then shows that every nilpotent endomorphism has trace zero; together with the nondegeneracy of the trace pairing, this makes the category semisimple, and a theorem of Deligne provides a faithful symmetric tensor functor to finite-dimensional super vector spaces. We then identify the resulting super tensor network with the Regts-Sevenster model exactly, viz. with its Eulerian-subgraph expansion and its sign of $-1$ for every fermionic circuit. An appendix gives an independent and direct proof of the nilpotent-trace step, showing that in a rigid symmetric $\mathbb{C}$-linear category with $\mathrm{End}(\mathbf{1})=\mathbb{C}$, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.

W. Whistler · 0 citations