Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
Non left-orderable surgeries on twisted torus knots
Kazuhiro Ichihara
Nihon University
College of Humanities and Sciences
Joint work withYuki Temma (Nihon Univ.)
MSJ Spring Meeting 2015 March 24, 2015
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Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
L-space Conjecture
L-space Conjecture [Boyer-Gordon-Watson, 2011]
M: an irreducible rational homology sphere
M is anL-space if and only if π1(M) is not LO
left-orderability
A non-trivial group Gis called left-orderable (LO) if ∃: a strict total order< onGwhich is left invariant:
g < h −→ f g < f h for ∀f, g, h∈G
L-space
A rational homology sphere M is called anL-space if rkHFd(M) =|H1(M;Z)|holds
Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
Left-orderable surgery and L-space surgery
K: a knot in 3-sphere S3
left-orderable surgery
Dehn surgery on K is called anon left-orderable surgery if it yields a 3-manifold with non left-orderable fund. group.
L-space surgery
Dehn surgery on K is called anL-space surgery if it yields a 3-manifold which is an L-space.
Question
Which knots inS3 have non-LO and/orL-space surgery?
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Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
Known results - Twisted Torus knots -
Twisted torus knotK(3,−4; 2,2)
Note:
twisted torus knotK(3,5; 2, s−2) = (−2,3,2s+ 1)-pretzel knot
Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
Known results - Twisted Torus knots -
Theorem [Vafaee, 2014]
For p≥2,k≥1,r >0and 0< s < p, K(p, kp±1;s, r) has anL-space surgery
if and only if either s=p−1 or s∈ {2, p−2} andr = 1.
Corollary
K(3, q; 2, s) has an L-space surgery if q >0and s≥1.
Theorem [Clay-Watson, 2013]
K(3,3k+ 2; 2, s) has a non left-orderable surgery if (1) k≥0and s= 1, or (2) k= 1 ands≥0.
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Non LO-surgeries K. Ichihara
Introduction L-space Conjecture Dehn surgery
Results Known Results Result
Result
Theorem [Ichihara-Temma]
For k, s≥0,K(3,3k+ 2; 2, s)has a non left-orderable surgery.
