Non-left-orderable surgeries and generalized Baumslag–Solitar relators

Kazuhiro Ichihara, Yuki Temma
2015 Journal of knot theory and its ramifications  
L-space vs left-orderable L-space Conjecture [Boyer-Gordon-Watson, 2011] M : an irreducible rational homology sphere M is an L-space if and only if π 1 (M ) is not LO L-space A rational homology sphere M is called an L-space if rk HF (M ) = |H 1 (M ; Z)| holds for HF (M ): Heegaard Floer homology. left-orderable group A non-trivial group G is called left-orderable (LO) if ∃ <: a strict total order on G which is left invariant: Left-orderable surgery and L-space surgery K: a knot in 3-sphere S 3
more » ... not in 3-sphere S 3 K(p/q): a 3-manifold obtained by Dehn surgery on K along the slope p/q left-orderable surgery A Dehn surgery on K is called a left-orderable surgery if it yields a closed 3-manifold with π 1 (K(p/q)) is left-orderable. L-space surgery A Dehn surgery on K is called an L-space surgery if it yields a closed 3-manifold which is an L-space. Known results -Twisted Torus knots - Theorem [Vafaee, 2014] For p ≥ 2, k ≥ 1, r > 0 and 0 < s < p, K(p, kp ± 1; s, r) has an L-space surgeries if and only if either s = p − 1 or s ∈ {2, p − 2} and r = 1. Corollary K(3, q; 2, s) has an L-space surgeries if q > 0 and s ≥ 1. / 15
doi:10.1142/s0218216515500030 fatcat:bnkmi6bk7bhwvhj4idcfgtzvjm