Bordered Heegaard Floer Homology and Knot Doubling Operators Adam Simon Levine Brandeis University Knot Concordance and Homology Cobordism Workshop Wesleyan University July 21, 2010 Adam Simon Levine Bordered HF and Knot Doubling Operators
Slice Knots and Links Definition A knot in S 3 is called topologically slice if it is the boundary of a locally flatly embedded disk in B 4 . smoothly slice if it is the boundary of a smoothly embedded disk in B 4 . A link is topologically/smoothly slice if it bounds a disjoint union of such disks. Adam Simon Levine Bordered HF and Knot Doubling Operators
Slice Knots and Links Definition A knot in S 3 is called topologically slice if it is the boundary of a locally flatly embedded disk in B 4 . smoothly slice if it is the boundary of a smoothly embedded disk in B 4 . A link is topologically/smoothly slice if it bounds a disjoint union of such disks. Big question: How do these two notions compare? Adam Simon Levine Bordered HF and Knot Doubling Operators
Whitehead and Bing Doubling Given a knot K , the positive Whitehead double, negative Whitehead double, and Bing double are: Wh + ( K ) Wh − ( K ) BD(K) Adam Simon Levine Bordered HF and Knot Doubling Operators
Whitehead and Bing Doubling Given a knot K , the positive Whitehead double, negative Whitehead double, and Bing double are: Wh + ( K ) Wh − ( K ) BD(K) We consider only untwisted doubles here. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles topologically slice? Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles topologically slice? Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles topologically slice? Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles topologically slice? Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes. It is equivalent to the four-dimensional surgery conjecture. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles topologically slice? Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes. It is equivalent to the four-dimensional surgery conjecture. Most people, including Freedman, think it’s not true. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles smoothly slice? Conjecture (Kirby’s problem list) Wh ± ( K ) is smoothly slice if and only if K is (smoothly) slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles smoothly slice? Conjecture (Kirby’s problem list) Wh ± ( K ) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph) If K is a strongly quasipositive knot different from the 1 unknot, then K is not smoothly slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles smoothly slice? Conjecture (Kirby’s problem list) Wh ± ( K ) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph) If K is a strongly quasipositive knot different from the 1 unknot, then K is not smoothly slice. If K is strongly quasipositive, then Wh + ( K ) is also strongly 2 quasipositive, hence not smoothly slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles smoothly slice? Conjecture (Kirby’s problem list) Wh ± ( K ) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph) If K is a strongly quasipositive knot different from the 1 unknot, then K is not smoothly slice. If K is strongly quasipositive, then Wh + ( K ) is also strongly 2 quasipositive, hence not smoothly slice. These were among the first known examples of knots that are topologically but not smoothly slice. (Akbulut, Gompf also found early examples.) Adam Simon Levine Bordered HF and Knot Doubling Operators
When are Whitehead doubles smoothly slice? Conjecture (Kirby’s problem list) Wh ± ( K ) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph) If K is a strongly quasipositive knot different from the 1 unknot, then K is not smoothly slice. If K is strongly quasipositive, then Wh + ( K ) is also strongly 2 quasipositive, hence not smoothly slice. These were among the first known examples of knots that are topologically but not smoothly slice. (Akbulut, Gompf also found early examples.) Bižaca used this to construct explicit examples of exotic smooth structures on R 4 . Adam Simon Levine Bordered HF and Knot Doubling Operators
The Ozsváth–Szabó invariant τ Knot Floer homology provides a knot invariant τ ( K ) ∈ Z , which vanishes for any smoothly slice knot. Adam Simon Levine Bordered HF and Knot Doubling Operators
The Ozsváth–Szabó invariant τ Knot Floer homology provides a knot invariant τ ( K ) ∈ Z , which vanishes for any smoothly slice knot. Theorem (Hedden) � 1 τ ( K ) > 0 τ ( Wh + ( K )) = 0 τ ( K ) ≤ 0 Adam Simon Levine Bordered HF and Knot Doubling Operators
The Ozsváth–Szabó invariant τ Knot Floer homology provides a knot invariant τ ( K ) ∈ Z , which vanishes for any smoothly slice knot. Theorem (Hedden) � 1 τ ( K ) > 0 τ ( Wh + ( K )) = 0 τ ( K ) ≤ 0 Corollary If K is any knot with τ ( K ) > 0 (e.g., any strongly quasipositive knot), then any iterated positive Whitehead double of K is not smoothly slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
Iterated Bing Doubling Any binary tree T specifies an iterated Bing double of K , denoted B T ( K ) . K Adam Simon Levine Bordered HF and Knot Doubling Operators
Iterated Bing Doubling Any binary tree T specifies an iterated Bing double of K , denoted B T ( K ) . K Adam Simon Levine Bordered HF and Knot Doubling Operators
Iterated Bing Doubling Any binary tree T specifies an iterated Bing double of K , denoted B T ( K ) . K Adam Simon Levine Bordered HF and Knot Doubling Operators
Generalized Borromean Rings The family of generalized Borromean links consists of all links obtained by taking iterated Bing doubles of the components of the Hopf link. Adam Simon Levine Bordered HF and Knot Doubling Operators
Main Theorem Are Whitehead doubles of iterated Bing doubles smoothly slice? Adam Simon Levine Bordered HF and Knot Doubling Operators
Main Theorem Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.) Let K be any knot with τ ( K ) > 0 (e.g., any strongly 1 quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of B T ( K ) is topologically but not smoothly slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
Main Theorem Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.) Let K be any knot with τ ( K ) > 0 (e.g., any strongly 1 quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of B T ( K ) is topologically but not smoothly slice. The all-positive Whitehead double of any generalized 2 Borromean link is not smoothly slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
Main Theorem Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.) Let K be any knot with τ ( K ) > 0 (e.g., any strongly 1 quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of B T ( K ) is topologically but not smoothly slice. The all-positive Whitehead double of any generalized 2 Borromean link is not smoothly slice. It is not known whether the links in (2) are topologically slice. Adam Simon Levine Bordered HF and Knot Doubling Operators
Doubling operators Given knots J , K and integers s , t , define the knot D J , s ( K , t ) = D K , t ( J , s ) as the boundary of the plumbing of an s -framed J -annulus and a t -framed K -annulus. J , s K , t Adam Simon Levine Bordered HF and Knot Doubling Operators
Recommend
More recommend
