I already asked this question on the Math Stack Exchange but did not get an answer.
I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be
found [here](http://www.mi.uni-koeln.de/%7Egeiges/contact05.pdf), and am trying to figure out the effect of a half Lutz twist on the Euler class
of the contact structure. During the proof of Propositioin 3.15, he basically claims that given a contact structure $\xi$
on $M$ and a transverse link $L$, then performing a half Lutz twist along the components
of $L$ yields a contact structure $\eta$ such that
$$e(\eta) = e(\xi) - 2PD[L]$$
where $PD: H\_1(M)\to H^2(M)$ is Poincaré-Duality.
I am however fairly certain, that I have proven the same result with $2PD[L]$ replaced
by $PD[L]$, here is my proof:


* Choose a tubular neighbourhood $V$ of $L$, such that on each component the contact structure is given by $\ker d\theta + r^2d\phi$.
* Start with a vector field $X$ tangent to $\xi$, which is in general position with the zero section,
does not vanish over $V$ and is given by $\partial\_r$ on the boundary of $V$, then $e(\xi) = PD[X\cap M]$.
* After the Lutz twist the $X$ I have in mind is not tangent to $\eta$, however we
can replace $X$ on $V$ by $\partial\_r$, such that $X$ now vanishes along $L$.
The new Euler class is thus given by
$$e(\eta) = PD[X\cap M] = e(\xi) + PD[L].$$


Can anyone spot a mistake in my argument, or is the statement in Geiges wrong?



Effect of a Lutz twist on Euler number
I have actually solved this by now, his statement is indeed correct but
the proof is a bit misleading. I will only discuss the case where $K$ is a knot.


Instead of choosing a field which does not vanish in a tubular neighbourhood of $K$, one simply chooses a generic vector field which is given by the radial vector field $\partial\_r$ in a tubular neighbourhood of $K$, which vanishes along $K$.


After the Lutz twist the same vector field is also tangent to $\eta$
and still vanishes along $K$, however, the Lutz twist has made $K$ a negatively transverse knot such that the difference $e(\eta) - e(\xi)$ is
given by $-2PD[K]$.


