Session S28 - Knots, Surfaces, 3-manifolds

Wednesday, July 14, 16:40 ~ 17:10 UTC-3

Taut foliations from double-diamond replacements

Rachel Roberts

Washington University in St Louis, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)$. We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if $R$ admits a double-diamond taut sutured manifold decomposition, then for every boundary slope except one, there is a co-oriented taut foliation of $M$ that intersects $\partial M$ transversely in a foliation by curves of that slope. In the case that $M$ is the complement of a knot $\kappa$ in $S^3$, the exceptional filling is the meridional one; in particular, restricting attention to rational slopes, it follows that every manifold obtained by non-trivial Dehn surgery along $\kappa$ admits a co-oriented taut foliation. As an application, we show that if $R$ is a Murasugi sum of surfaces $R_1$ and $R_2$, where $R_2$ is an unknotted band with an even number $2m\ge 4$ of half-twists, then every manifold obtained by non-trivial surgery on $\kappa= \partial R$ admits a co-oriented taut foliation.

Joint work with Charles Delman (Eastern Illinois University).

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