# Finding a ribbon graph for a mapping class group action

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.

1. 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 $M(g)$.

2. 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 $M(g)$.

Question

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?

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Dear Link S. It seems that you are the same person as "Link" and "Knot". Could you consolidate your MO accounts? Also, you should explain how ribbon graphs relate to $M(g)$, since not everybody is as familiar with Turaev's book as you are. – Misha Mar 11 '12 at 23:38
Duplicate of mathoverflow.net/questions/87567/… – Daniel Moskovich Mar 11 '12 at 23:38
@Misha, I will register. @ Daniel, it is not the same question as in your link. Or do you mean I need to study kno theory first as this is the answer to that question? – Link S Mar 12 '12 at 1:32