The triconnected Kontsevich graph complex

TL;DR

The paper proves that the tri-connected subcomplex $\mathsf{G}_n^{tri}$ computes the Kontsevich graph cohomology $H(\mathsf{G}_n)$, extending the known biconnected quasi-isomorphism. It develops an SPQR-tree filtration on $\mathsf{G}_n^{bi}$ and analyzes the associated graded to show the kernel is acyclic, thereby establishing the quasi-isomorphism $\mathsf{G}_n \simeq \mathsf{G}_n^{tri}$. This yields a smaller computational model and, for even $n$, provides improved lower bounds on cohomology via a loop-order spectral sequence, connecting graph cohomology with Lie-theoretic structures such as $\mathfrak{grt}_1$. The results have practical computational benefits and deepen the interplay between graph complexes and deformation theory.

Abstract

We show that a smaller version of the Kontsevich graph complex spanned by triconnected graphs is quasi-isomorphic to the full Kontsevich graph complex.

Figures (3)

• Figure 1: A biconnected graph (left) and its associated SPQR tree (right). The graph inscribed into a vertex of the SPQR tree is the skeleton of that vertex. The real edges are drawn as solid edges, and the virtual edges as dashed edges.
• Figure 2: Table of the dimensions of the 10-loop part of the three complexes $\mathsf{G}_0$, $\mathsf{G}_0^{bi}$ and $\mathsf{G}_0^{bi}$ for various numbers of vertices $k$. The percentage is the fraction of the size of $\mathsf{G}_0$.
• Figure 3: The table shows $\mathop{\mathrm{dim}}\nolimits H^k(\mathsf{GC}_2^{g\text{-loop}})$ as computed in BW. There is no cohomology in negative degrees for any $g$, and no cohomology in degrees $k>g-3$, as indicated by the diagonal line.

Theorems & Definitions (10)

• Theorem 1.1
• Lemma 2.3
• proof
• Theorem 4.3: KWZ1
• Corollary 4.4
• Corollary 4.5
• Proposition 4.6: Z
• Lemma 4.7
• proof
• proof : Proof of Corollary \ref{['cor:sseq']}