Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is defined as follows.
Consider the cylinder $\Sigma \times [0, 1]$. The top and bottom parametrizations are the identity and $g$ respectively, where $g$ is an element of a mapping class group of $\Sigma$. Denote this cobordism by $M(g)$.
Define an action of $g$ by $\epsilon(g)=\tau(M(g))$.
To calculate $\tau(M(g))$, we first need to find a special ribbon graph which presents $M(g)$.
Is there any algorthm to find such a special ribbon graph if we are given an element $g$ of a mapping class group?
More generally, is there any way to find a special ribbon graph that presents a given cobordism?
Could you give me references?