All: I'm reading a paper that makes mention of the twisting $tw (\gamma,S) $ , where $\gamma$ is a simple, closed Legendrian curve in a surface $S$ , and $S$ is embedded in a contact 3-manifold $(M,\theta)$; $\theta$ is a contact structure on $M$; unfortunately I don't have a link. I would like to know the actual definition of this $tw$. For the sake of context, the paper states that both $\theta$ and $S$ give $\gamma$ a framing, i.e. a trivialization of its normal bundle by taking a vector field X normal to $\gamma$ and tangent to $\theta$ or $S$, respectively; then $tw(\gamma,S)$ measures how many times the vector field X corresponding to $\theta$ rotates as $\gamma$ is traversed measured with respect to the vector field corresponding to $S$.
I'm hoping someone can tell me or refer me to a formal definition for this $tw$. Thanks in advance.
The "twisting number" is a relative Thurston-Bennequin invariant with respect to a framing. It is the integer difference between the number of twists in the normal framing and the number of twists in the given framing F, with clockwise twists counting as 1 and counterclockwise as -1.
The precise definition may be found on e.g. http://www-bcf.usc.edu/~khonda/math599/notes.pdf Page 8.