History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Meditation on generalized Property R
with Bob Gompf and Abby Thompson
Dublin
August 2015
1/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Theorem (Gabai 1987)
If surgery on a knot K ⊂ S 3 gives S 1 × S 2 , then K is the unknot.
Question: If surgery on a link L of n components gives
#n (S 1 × S 2 ), what is L?
Homology argument shows that
each pair of components in L is algebraically unlinked and
the surgery framing on each component of L is the 0-framing.
2/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Conjecture (Naive)
If surgery on a link L of n components gives #n (S 1 × S 2 ), then L
is the unlink.
Why naive?
The result of surgery is unchanged when one component of L is
replaced by a band-sum to another. So here’s a counterexample:
3/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
The 4-dimensional view of the band-sum operation:
Integral surgery on L ⊂ S 3 ↔ 2-handle addition to ∂B 4 .
Band-sum operation corresponds to a 2-handle slide
U'
V'
U
V
Effect on dual handles: U slid over V ↔ V 0 slid over U 0 .
4/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
The fallback:
Conjecture (Generalized Property R)
If surgery on an n component link L ⊂ S 3 gives #n (S 1 × S 2 ),
then, perhaps after some handle-slides, L becomes the unlink.
Conjecture is unknown even for n = 2.
Questions: If it’s not true, what’s the simplest counterexample?
What’s the simplest knot that could be part of a counterexample?
It must be slice in some homotopy 4-ball:
S
3
L
3-handles
L
2-handles
5/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Question: Can the unknot be part of a counterexample?
Answer: No
Theorem
If U is the unknot and surgery on L = U ∪ V gives #2 (S 1 × S 2 ),
then some handle-slides of V over U change L to the unlink.
Idea: Since U is the unknot, 0-surgery on U gives S 1 × S 2 ⊃ V ,
and S 1 × S 2 ⊃ the dual knot U 0 = S 1 × (point).
U'
V
6/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
This means that two different fillings of η(V ) (with and without
surgery) give reducible manifolds. Deep theorem of Gabai says this
means V lies in a ball in S 1 × S 2 . Can get it into a ball in the
complement of U 0 by letting V cross U 0 , i. e. V is handle-slide
over U. Once inside a ball then Gabai’s Property R means V is the
unknot, so L becomes the unlink.
U'
V
7/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Can ramp this argument up, to consider what is the simplest knot
that could occur in a 2-component counterexample.
Theorem
If U is a fibered knot and surgery on L = U ∪ V gives
#2 (S 1 × S 2 ), then a sequence of handle-slides (back and forth)
gives a link with a component of genus lower than U.
F
Idea: Since U is fibered, surgery on
U gives a fibered manifold M ⊃
U 0 ∪ V with fiber F . Surgery on
V in M gives #2 (S 1 × S 2 ).
I
V
h
U'
8/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Theorem (S-Thompson 2008)
If surgery on V in fibered M gives a reducible manifold, then
[either V lies in a ball, or V is cabled, or] V lies in a fiber.
So here V lies in F after passing through U 0 , i. e. after V is
changed by handle-sliding over U
multiple times
U'
V
9/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
What 0-framed surgery on V ⊂ M now looks like: boundary union
of two compression-bodies, each ∂ + = F and ∂ − = F 0 =, the
surface obtained from F by compressing along V .
F
U'
V'
F'
h
U'
1
2
#2(S x S )
10/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Then two handle-slides of V 0 over U 0 makes V 0 disjoint from F 0 .
l
two handle-slides of U over V gives U the Seifert surface F 0
F
F
V'
F'
U'
1
V'
2
#2(S x S )
F'
U'
Corollary
Both trefoil and figure 8 knots do not appear in a counterexample
But we knew that before: neither is slice in a homotopy 4-ball.
11/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Genus 2 fibered knots: Can any be part of a 2-component
counterexample? Has to be slice in a homotopy 4-sphere.
Maybe the simplest: Q = K # − K = squareknot, K the trefoil.
Examples of links L = Q ∪ V so surgery gives #2 (S 1 × S 2 ):
Know from earlier that none of these are counterexamples.
12/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
The square knot is so simple, can completely characterize (up to
handle-slides over Q) all V so surgery on Q ∪ V gives
#2 (S 1 × S 2 ). In fact {such V } ↔ Q.
Proof uses specific information about monodromy of Q and
Heegaard analysis:
F'
1
U'
2
#2(S x S )
F
F'
Reasonable idea: Figure out a way to show that each possible V is
not a counterexample.
13/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
n strands
The Akbulut-Gompf example again
n strands
+1
-1
Theorem (Akbulut-Gompf 1991)
If this handle description of a homotopy 4-sphere can be handle-slid
to standard, then this presentation is Andrews-Curtis trivial
< a, b | aba = bab, an+1 = b n >
14/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Monodromy of fiber F (genus two surface) easy to describe –
rotate and flip:
Quotient of rotation is 4-punctured sphere.
15/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Fiber of Q is 3-fold branched cover of 4-punctured sphere
In 4-punctured sphere, essential curves classified by slope ∈ Q.
Heegaard analysis shows that to get S 2 × S 1 #S 2 × S 1 , curve V
must be homeomorphic lift of curve with slope qp , q odd.
(Example here is slope 49 , turned on side) .
16/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
The challenge: Find unspecified slides of V over Q that turns
n strands
-1
n strands
+1
into
It won’t be easy: note the n-strand sections have non-trivial linking
number with Q. And what’s with the ±1 twists?
17/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Why do this?
“Natural” position of V ∪ Q might be better for either
a) detecting useful slides of Q over V , sliding to the unlink or
b) finding an obstruction to sliding to the unlink.
A geometric resolution of Andrews-Curtis counterexample?
Sutured manifold test: can V really be put on a fiber?
Step 1: Push whole apparatus near fiber:
18/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Step 2: Seek good picture of the monodromy:
Zeeman on the trefoil knot
“I personally found it hard to visualise how the complement of a
knot could be fibered so beautifully, until I heard a talk by John
Stallings on Neuwirth knots”
Idea: Identify opposite sides of a hexagon, get a torus.
Monodromy of trefoil knot is π/3 rotation.
Trefoil knot monodromy.
Filled in square knot.
19/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
(If square knot not filled in, boundary must remain fixed:)
Step 3: Visualize apparatus:
-1/2
1/2
Book
kooB
1/2
-1/2
Initial position
Signs of 12 -twist change, cancel
20/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
n-strand bands now appear flat, but change level at green dots.
Knot boundary filled in with disk:
Begin to peel
lift brown arc full level,
applying monodromy
Continue n times
21/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Example: n = 4
In monodromy picture
Back in S 3
22/23
History and motivation
Ruling out counterexamples
Importing Heegaard theory
The Akbulut-Gompf example again
Let’s classify!
Another picture of the
branched covering
Answer:
n
2n+1
23/23