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Taut foliations, positive 3‐braids, and the L‐space conjecture
Journal of Topology  (IF1.582),  Pub Date : 2020-05-02, DOI: 10.1112/topo.12147
Siddhi Krishna

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 $r < 2 g ( K ) − 1$ 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 $r < g ( K )$.