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 S3, 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 S3, 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.

L-spaces, taut foliations and the Whitehead link

“…Taut foliations on surgeries on knots are constructed, for example, in [Rob01], [LR14], [DR19], [DR20], [Kri20] and it is possible to prove the left orderability of some of these manifolds by determining which of these foliations have vanishing Euler class, as done in [Hu19]. Another approach to study the left orderability of surgeries on knots is via representation theoretic methods, as presented in [CD18].…”