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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  )   .  This paper relies extensively on color figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.

Taut foliations, positive 3‐braids, and the L‐space conjecture