Let $A_\Gamma$ be an Artin group. The Salvetti complex $Sal(A_{\Gamma})$ can be briefly defined as the $2$-presentation complex associated to the usual presentation of the Artin group after attaching higher dimensional cells for each complete subgraph of $\Gamma$.
Obviously the $0$-skeleton is given by a unique cell $\sigma_\emptyset$, its $1$-skeleton is given by $1$-cells $\sigma_v$, one for each vertex of $\Gamma$ and its $2$-skeleton is given by $2$ cells $\sigma_e$ one for each edge of $\Gamma$.
I'm trying to explicitly compute the associated chain complex. It is evident that $C_n(Sal(A_{\Gamma}))=\bigoplus\limits_{X\subset I_n}\mathbb{Z}\sigma_X$, where $I_n$ is the set of complete subgraphs of $n$-vertices of $\Gamma$. However, I can't explicitly compute the boundary maps.
Given an ordering on the vertices of $\Gamma$ we can set an orientation and one can trivially see the following facts:
- $\partial(\sigma_v)=0$ for every vertex $v$
- $\partial(\sigma_e)=0$ if $e$ is an edge labelled with an even number
- $\partial(\sigma_e)=\sigma_u-\sigma_v$ if $e=\lbrace u,v\rbrace$ is an edge labelled with an odd number. The sign depends on the orientation of the vertices of $\Gamma$.
However, I can't see how to compute the image of a cell $\sigma_X$ for $X$ a complete subgraph of $n\geq 3$ vertices. If there is not a general formula for that an example might work as an answer, (for example if we consider $\Gamma$ to be the complete graph with $3$ vertices and whose edges are labelled with a $2,3$ and $4$, what is the image of the $3$-cell?)
I've read that the differential of the Salvetti complex is zero in the case where $A_\Gamma$ is even (all edges of $\Gamma$ are labelled with even numbers) and I guess that this will be easy to prove once I understand how the differential works.
Thanks for your help.
