Title : Producing Klein bottles by two distinct Dehn fillings
(a joint work with Nabil Sayari)
Speaker : Daniel Matignon (Universite de Provence)
Abstract :

Let $M$ be a compact, connected and orientable $3$-manifold whose boundary contains a $2$-torus $T$. Suppose that $M$ is hyperbolic, and there exist two distinct simple closed curves $\gamma_1, \gamma_2$ on $T$ such that gluing along a solid torus $V_i$ (for $i=1$ or $2$) in such a way that the boundary of a meridian disk of $V_i$ is attached on $\gamma_i$, produce $3$-manifolds, both containing a Klein bottle (in particular, not hyperbolic). We give a bound on the distance $\Delta(\gamma_1, \gamma_2)$, i.e. the minimal geometric intersection number between $\gamma_1$ and $\gamma_2$ after isotopy. We show that generically $\Delta(\gamma_1, \gamma_2)\leq 4$. More precisely, there are exactly two cases for which $\Delta(\gamma_1, \gamma_2)>5$~: either $\Delta(\gamma_1, \gamma_2)=6$ and so $M$ is the $3$-manifold obtained by a $2/1$-Dehn filling on the exterior of the Whitehead link in $S^3$, or $\Delta(\gamma_1, \gamma_2)=8$ and so $M$ is either the exterior of the figure-eight knot or the $3$-manifold obtained by a $-5/1$-Dehn filling on the exterior of the Whitehead link in $S^3$. Moreover, there exists at most one manifold $M$ for which $\Delta(\gamma_1,\gamma_2)=5$.

pdf-file