Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

Abstract

We construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 3-braid knot K in S 3 , where r < 2g(K) − 1 and g(K) denotes the Seifert genus of K. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot P (−2, 3, 7), and indeed along every pretzel knot P (−2, 3, q), for q a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 1-bridge braid in S 3 , where r < g(K).

Euler class of taut foliations and Dehn filling

“…Since b 1 (X) > 0, essential surfaces contained in X often serve as a convenient starting point for sutured manifold decomposition [Gab83,Gab87] and for building branched surfaces. As a result, taut foliations on manifolds obtained by Dehn filling are usually constructed by first building one on X and then extending it to the entire closed manifold using either meridional disks or stacking chairs (see [Rob95,Rob01a,Rob01b,LR14,Kri18,DR19a,DR19b] for instance). These foliations are transverse to the core of N .…”