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I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be found here, 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?

Universal Property
  • 301

2 Answers2

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(Sorry this is more of a comment, but I can't seem to comment) I forget the details of the Lutz twist, but I'm guessing $X$ vanishes along $L$ \emph{with multiplicity 2}.

no_idea
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I can't quite remember the details of what was wrong with my argument, but at least one of the objects I use does not exist. The reason why Geiges statement is correct is because the half Lutz twist reverses the orientation of the contact structure along L turning it from a positively transverse knot into a negatively transverse one.

Universal Property
  • 301