A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Montesinos knots, Hopf plumbings and L-space surgeries Kenneth Baker ‡ Allison Moore † † Rice University ‡ University of Miami October 24, 2014 Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions A longstanding question Which knots admit lens space surgeries? V 1971 (Moser) µ V X 1977 (Bailey-Rolfsen) 1980 (Fintushel-Stern) 1990 (Berge) α µ α = p µ + q λ Cyclic Surgery Theorem (CGLS) + Berge’s construction = “The Berge Conjecture.” Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
� A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions L-spaces (Ozsv´ ath-Szab´ o, Rasmussen): Knot Floer homology. � · · · ⊂ F i − 1 C ⊂ F i C ⊂ . . . K ⊂ Y H ∗ ( F i C / F i − 1 C ) = HFK( K ) = � � m , s � HFK m ( S 3 , K , s ) . ∆ K ( t ) = � s χ ( � HFK( K , s )) · t s A Q HS 3 Y is an L-space if | H 1 ( Y ; Z ) | = rank � HF ( Y ). Ex: S 3 , all lens spaces, 3-manifolds with finite π 1 . Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Motivating question revisited Question Which knots admit lens space surgeries? becomes Question Which knots admit L-space surgeries? Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions L-space surgery obstructions Theorem (Ozsv´ ath-Szab´ o) If K admits an L-space surgery, then for all s ∈ Z , HFK ( K , s ) ∼ � = F or 0 (and some other conditions on Maslov grading). Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions L-space surgery obstructions Theorem (Ozsv´ ath-Szab´ o) If K admits an L-space surgery, then for all s ∈ Z , HFK ( K , s ) ∼ � = F or 0 (and some other conditions on Maslov grading). Corollary (Determinant-genus inequality) If det ( K ) > 2 g ( K ) + 1 , then K is not an L-space knot. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions L-space surgery obstructions Theorem (Ozsv´ ath-Szab´ o) If K admits an L-space surgery, then for all s ∈ Z , HFK ( K , s ) ∼ � = F or 0 (and some other conditions on Maslov grading). Corollary (Determinant-genus inequality) If det ( K ) > 2 g ( K ) + 1 , then K is not an L-space knot. Proof. If K is an L-space knot, then | a s | ≤ 1 ∀ coefficients a s of ∆ K ( t ). Then, � det( K ) = | ∆ K ( − 1) | ≤ | a s | ≤ 2 g ( K ) + 1 . s Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions More geometric obstructions Theorem (Ni, Ghiggini) HFK( K , g ( K )) ∼ � = F . K is fibered if and only if Thus L -space knots are fibered. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions More geometric obstructions Theorem (Ni, Ghiggini) HFK( K , g ( K )) ∼ � = F . K is fibered if and only if Thus L -space knots are fibered. Theorem (Hedden) An L-space knot K supports the tight contact structure; equivalently, an L-space knot is strongly quasipositive. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Classification theorem Theorem (Baker-M.) Among the Montesinos knots, the only L-space knots are the pretzel knots P ( − 2 , 3 , 2 n + 1) for n ≥ 0 , and the torus knots T (2 , 2 n + 1) for n ≥ 0 . Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Montesinos knots � β 1 � , β 2 α 2 , . . . , β r K = M | e α 1 α r Figure: M ( 3 4 , − 2 5 , 1 3 | 3). Where α i , β i , e ∈ Z and α i > 1, | β i | < α i , and gcd( α i , β i ) = 1. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Ingredients for proof We need only consider fibered, non-alternating Montesinos knots, � β 1 � , β 2 α 2 , . . . , β r K = M | e α 1 α r and we assume r ≥ 3, because r ≤ 2 implies K is a two-bridge link. Theorem (Ozsv´ ath-Szab´ o) An alternating knot admits an L-space surgery if and only if K ≃ T (2 , 2 n + 1) , some n ∈ Z . Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Fibered Montesinos knots (Hirasawa-Murasugi): Classified fibered Montesinos knots with � � α 1 , β 2 β 1 α 2 , . . . , β r α r | e their fibers. For K = M , 1 β i = α i 1 x 1 − 1 x 2 − ... − 1 x m S i := [ x 1 , . . . , x m ] have two cases of S i : 1 α i are all odd � strict continued fractions. 2 α 1 is even, α i is odd for i > 1 � even continued fractions. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Example: odd case Each β i /α i has a strict continued fraction: S i = [2 a ( i ) 1 , b ( i ) 1 , . . . , 2 a ( i ) q i , b ( i ) q i ] Hirasawa-Mursagi give strong restrictions on e , S 1 , . . . , S m when M is fibered. Figure: Image of odd-type Seifert surface borrowed from Hirasawa-Murasugi. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Open books for three-manifolds ( F , φ ) —an open book for closed 3-manifold Y . L = ∂ F is the binding. F is the fiber surface. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Open books for three-manifolds ( F , φ ) —an open book for closed 3-manifold Y . L = ∂ F is the binding. F is the fiber surface. ξ —a contact structure on Y . Locally, ker α , α ∧ d α � = 0 (Thurston-Winkelnkemper - 1975) Every ( F , φ ) induces a contact structure. (Giroux - 2000) { or. ξ on Y } / isotopy ← → { ( F , φ ) for Y } / positive stabilization Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Plumbings of Hopf bands Hopf links: L + = { ( z 1 , z 2 ) ∈ S 3 ⊂ C 2 | z 1 z 2 = 0 } . L − = { ( z 1 , z 2 ) ∈ S 3 ⊂ C 2 | z 1 z 2 = 0 } . Pos/neg (de)stabilization ↔ (de)plumbing of pos/neg Hopf bands. Figure: The connected sum of a positive and negative Hopf band. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Lemma (Contact Structures Lemma) 1 (Goodman): If F ⊃ H − , then ξ ( F ,φ ) is overtwisted. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Lemma (Contact Structures Lemma) 1 (Goodman): If F ⊃ H − , then ξ ( F ,φ ) is overtwisted. 2 (Yamamoto): If F contains a twisting loop, then ξ ( F ,φ ) is overtwisted. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Lemma (Contact Structures Lemma) 1 (Goodman): If F ⊃ H − , then ξ ( F ,φ ) is overtwisted. 2 (Yamamoto): If F contains a twisting loop, then ξ ( F ,φ ) is overtwisted. 3 (Giroux): If F ⊃ H + and ( F , φ ) = ( F ′ , φ ′ ) ∗ ( H + , π + ) then ξ ( F ,φ ) ≃ ξ ( F ′ ,φ ′ ) . Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Theorem (Baker-M.) A fibered Montesinos knot that supports the tight contact structure is isotopic to either Figure: Left: odd type. Right: even type. and its fiber is obtained from the disk by a sequence of Hopf plumbings. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Odd case Repeatedly apply the Contact Structures Lemma, parts 1 & 2 to identify negative Hopf bands and/or twisting loops. Cull these knots because they support an Figure: Finding negative Hopf bands in F . overtwisted contact structure. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
A Question Surgery obstructions Montesinos knots Hopf plumbings More Questions Odd case Odd fibered Montesinos knots without a H − remain. Successively deplumb H + until a single H + remains. These knots support the tight contact structure. Allison Moore Montesinos knots, Hopf plumbings and L-space surgeries
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