Alternating Knots & Montesinos Knots Satisfy the (Classical) L-space Surgery Conjecture Charles Delman Joint work with Rachel Roberts Background Foliations Heegaard-Floer Homology Knots & Surgery Conjectures Results & Methods Results Methods Alternating Knots & Montesinos Knots Satisfy the (Classical) L-space Surgery Conjecture Charles Delman Joint work with Rachel Roberts March 28, 2017 GEAR Seminar, UIUC
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AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Alternating Knots & Montesinos Knots Satisfythe (Classical) L-space Surgery Conjecture
A foliation is a decomposition of a manifold into leaves of lowerdimension. Locally, we have charts Rm × Rn, with transitionsthat preserve the horizontal levels Rm × {y}.
Leaf dimension
Co-dimension
We consider foliations of smooth 3-manifolds with2-dimensional C 1-embedded leaves (co-dimension 1).
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Taut Foliations
Definition
A co-dimension 1 foliation of a 3-manifold is taut if there is acircle transversely intersecting every leaf.
Remark: A closed manifold admitting a taut foliation isuniversally covered by R3, hence is irreducible and has infinitefundamental group.
Definition
A 3-manifold is foliar if it admits a taut, co-orientable(co-dimension 1) foliation.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Heegaard-Floer Homology
An homology theory for rational homology 3-spheres.
Introduced by P. Ozsvath & Z. Szabo.
HF (M) is a vector space over F2.
Rank(HF (M)) ≥ |H1(M,Z)|.If equality holds, M is an L-space.
M admits a taut, co-orientable foliation ⇒ M is not an L-space
Does the converse hold for irreducible 3-manifolds?(Ozsvath–Szabo, Boyer-Gordon-Watson, Juhasz?)
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Knots
A (classical) knot is an (n − 2)-sphere embedded in ann-sphere, in particular, for n = 3.
Knot in S 3
(alternating)
Note that a regular neighborhood (“fattening up”) of a knot isa solid torus.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Dehn Surgery
Remove a solid torus (a “fattened up” knot) from S3 andglue in a solid torus by a homeomorphism of T 2.
The result depends only on the curve to which themeridian is glued.
l longitudes and m meridians, l ,m relatively prime, giveDehn surgery coefficient m
l ∈ Q ∪ 10 .
Coefficient 1/0 is trivial surgery (yielding S3 back).
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Two Interesting Types of Knots
In particular, we consider two classes of knots:
Alternating knotsMontesinos knots:
M(1/3, 2/5, 3/5, -1)
The pretzel knots are a subset of the Montesinos knots:
(3,3,3)-Pretzel Knot
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Terminology
Definition
A knot k is persistently foliar if every manifold obtained bynon-trivial Dehn surgery on k is foliar.
Definition
A knot k is an L-space knot if some non-trivial surgery on kyields an L-space.
Corollary
If a knot is persistently foliar, it is not an L-space knot.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Conjectures [D-Roberts]
Restricting attention to surgery on knots k ⊂ S3, we conjecturethe following:
L-space Knot Conjecture If k does not admit a non-trivialreducible or L-space surgery, then k is persistently foliar.
More generally,
L-space Surgery Conjecture A manifold obtained by Dehnsurgery on k is foliar if and only if it is irreducible and not anL-space.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Results
Theorem (D-Roberts)
All alternating knots satisfy the L-space surgery conjecture. Inparticular, every non-torus alternating knot is persistently foliar.
Remark: For torus knots, the result follows from theclassification of their foliar (Boyer, Eisenbud-Hirsch-Neumann,Jenkins-Neumann, Raimi) and L-space (Hedden) surgeries.
Theorem (D-Roberts)
All Montesinos knots satisfy the L-space surgery conjecture. Inparticular, every Montesinos knot that is not an L-space knot ispersistently foliar.
Remark: The result for L-space knots follows from work ofBaker, Lidman, Hedden, Moore, and Roberts.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Finite Depth Spines
Build a spine (Casler) from a finite succession oftransversely intersecting surfaces.
Locally:
Surfaceneighborhood
Double pointneighborhood
Triple pointneighborhood
p pp
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Smoothing Instructions
Successively introduce smoothing instructions at singularpoints to obtain a branched surface (continuous tangentplane field):
Surfaceneighborhood
Double pointneighborhood
Triple pointneighborhood
pp
p
Eventually obtain a transversely orientable laminarbranched surface for which the complement of an I -bundleneighborhood is a taut sutured manifold.
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
I -bundle Neighborhood
Surfaceneighborhood
Double pointneighborhood
Triple pointneighborhood
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Notation
Arrow-diamond notation at a double point with onedistinguished sector:
=
=
=
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Smoothings at a Triple Point
There are 12 possible smoothings at a triple point:
=
=
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Work in the Knot Exterior
Work in the knot exterior: S3 \ KIntroduce a “tube” around K : T = ∂N(K ) ⊂ S3 \ KT is part of the spine.
Convention: Outward normal to T points into knotcomplement, out of N(K ).
T
K
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Meridional Cusps → Persistence
Goal:
Build spine having meridional intersections with T .
Smooth to branched surface Σ with even (> 0) number ofmeridional branch curves with outward sink direction on T .
After any rational Dehn surgery, these yield an evennumber of longitudinal sutures, so a meridional disk fullydecomposes N(K ′) (as a taut sutured manifold).
T T
K'KDehn
surgery
AlternatingKnots &
MontesinosKnots Satisfythe (Classical)
L-spaceSurgery
Conjecture
CharlesDelman
Joint workwith RachelRoberts
Background
Foliations
Heegaard-FloerHomology
Knots & Surgery
Conjectures
Results &Methods
Results
Methods
Meridional Cusps → Persistence (continued)
Thus, as long as the other components of N(Σ)c are tautsutured manifolds, we obtain a taut co-orientable foliationin every manifold produced by (non-trivial) surgery.