Casson towers and filtrations of the smooth knot concordance group Arunima Ray AMS Central Sectional Meeting Washington University at St. Louis St. Louis, Missouri October 19, 2013
Introduction Goal Casson towers Results Definitions Definition A knot is slice if it bounds a smoothly embedded disk ∆ in B 4 . K ∆ S 3 B 4 Knots, modulo slice knots, form the smooth knot concordance group , denoted C .
Introduction Goal Casson towers Results Definitions Definition A knot is slice if it bounds a smoothly embedded disk ∆ in B 4 . K ∆ S 3 B 4 Knots, modulo slice knots, form the smooth knot concordance group , denoted C . There exist infinitely many smooth concordance classes of topologically slice knots (Endo, Gompf, Hedden–Kirk, Hedden–Livingston–Ruberman, Hom, etc.)
Introduction Goal Casson towers Results Approximating sliceness K ∆ S 3 B 4 A knot is slice if it bounds a disk in B 4 .
Introduction Goal Casson towers Results Approximating sliceness K ∆ S 3 B 4 A knot is slice if it bounds a disk in B 4 . Two ways to approximate sliceness: • knots which bound disks in [[approximations of B 4 ]]
Introduction Goal Casson towers Results Approximating sliceness K ∆ S 3 B 4 A knot is slice if it bounds a disk in B 4 . Two ways to approximate sliceness: • knots which bound disks in [[approximations of B 4 ]] • knots which bound [[approximations of disks]] in B 4
Introduction Goal Casson towers Results The n –solvable filtration of C Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0 , a knot K is in F n (and is said to be n –solvable ) if K bounds a smooth, embedded disk ∆ in [[an approximation of B 4 ]]
Introduction Goal Casson towers Results The n –solvable filtration of C Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0 , a knot K is in F n (and is said to be n –solvable ) if K bounds a smooth, embedded disk ∆ in a smooth, compact, oriented 4–manifold V with ∂V = S 3 such that • H 1 ( V ) = 0 , • there exist surfaces { L 1 , D 1 , L 2 , D 2 , · · · , L k , D k } embedded in V − ∆ which generate H 2 ( V ) and with respect to which the intersection form is � � 0 1 � , 1 0 • π 1 ( L i ) ⊆ π 1 ( V − ∆) ( n ) for all i , • π 1 ( D i ) ⊆ π 1 ( V − ∆) ( n ) for all i .
Introduction Goal Casson towers Results The n –solvable filtration of C Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0 , a knot K is in F n (and is said to be n –solvable ) if K bounds a smooth, embedded disk ∆ in a smooth, compact, oriented 4–manifold V with ∂V = S 3 such that • H 1 ( V ) = 0 , • there exist surfaces { L 1 , D 1 , L 2 , D 2 , · · · , L k , D k } embedded in V − ∆ which generate H 2 ( V ) and with respect to which the intersection form is � � 0 1 � , 1 0 • π 1 ( L i ) ⊆ π 1 ( V − ∆) ( n ) for all i , • π 1 ( D i ) ⊆ π 1 ( V − ∆) ( n ) for all i . Clearly, · · · ⊆ F n ⊆ F n − 1 ⊆ · · · ⊆ F 0 ⊆ C
Introduction Goal Casson towers Results The n –solvable filtration of C • F 0 = { K | Arf ( K ) = 0 } • F 1 ⊆ { K | K is algebraically slice } • F 2 ⊆ { K | various Casson–Gordon obstructions vanish }
Introduction Goal Casson towers Results The n –solvable filtration of C • F 0 = { K | Arf ( K ) = 0 } • F 1 ⊆ { K | K is algebraically slice } • F 2 ⊆ { K | various Casson–Gordon obstructions vanish } • ∀ n, Z ∞ ⊆ F n / F n +1
Introduction Goal Casson towers Results The grope filtration of C Definition For any n ≥ 1 , a knot K is in G n if K bounds a grope of height n in B 4 .
Introduction Goal Casson towers Results The grope filtration of C Definition For any n ≥ 1 , a knot K is in G n if K bounds a grope of height n in B 4 . K
Introduction Goal Casson towers Results The grope filtration of C Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0 , G n +2 ⊆ F n
Introduction Goal Casson towers Results Topologically slice knots Let T denote the set of all topologically slice knots. ∞ � T ⊆ F n n =0
Introduction Goal Casson towers Results Topologically slice knots Let T denote the set of all topologically slice knots. ∞ � T ⊆ F n n =0 How can we use filtrations to study smooth concordance classes of topologically slice knots?
Introduction Goal Casson towers Results Positive and negative filtrations of C Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0 , a knot K is in P n (and is said to be n –positive ) if K bounds a smooth, embedded disk ∆ in [[an approximation of B 4 ]]
Introduction Goal Casson towers Results Positive and negative filtrations of C Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0 , a knot K is in P n (and is said to be n –positive ) if K bounds a smooth, embedded disk ∆ in a smooth, compact, oriented 4–manifold V with ∂V = S 3 such that • π 1 ( V ) = 0 , • there exist surfaces { S i } embedded in V − ∆ which generate H 2 ( V ) and with respect to which the intersection form is � � � 1 , • π 1 ( S i ) ⊆ π 1 ( V − ∆) ( n ) for all i ,
Introduction Goal Casson towers Results Positive and negative filtrations of C Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0 , a knot K is in N n (and is said to be n –negative ) if K bounds a smooth, embedded disk ∆ in a smooth, compact, oriented 4–manifold V with ∂V = S 3 such that • π 1 ( V ) = 0 , • there exist surfaces { S i } embedded in V − ∆ which generate H 2 ( V ) and with respect to which the intersection form is � � − 1 � , • π 1 ( S i ) ⊆ π 1 ( V − ∆) ( n ) for all i ,
Introduction Goal Casson towers Results Positive and negative filtrations of C Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0 , a knot K is in N n (and is said to be n –negative ) if K bounds a smooth, embedded disk ∆ in a smooth, compact, oriented 4–manifold V with ∂V = S 3 such that • π 1 ( V ) = 0 , • there exist surfaces { S i } embedded in V − ∆ which generate H 2 ( V ) and with respect to which the intersection form is � � − 1 � , • π 1 ( S i ) ⊆ π 1 ( V − ∆) ( n ) for all i , These filtrations can be used to distinguish smooth concordance classes of topologically slice knots
Introduction Goal Casson towers Results Goal Prove a version of the result relating the grope filtration and n –solvable filtration, for the positive/negative filtrations
Introduction Goal Casson towers Results Casson towers Any knot bounds a kinky disk in B 4 , i.e. a disk with transverse self-intersections.
Introduction Goal Casson towers Results Casson towers Any knot bounds a kinky disk in B 4 , i.e. a disk with transverse self-intersections. Any knot which bounds such a kinky disk with only positive self-intersections lies in P 0 .
Introduction Goal Casson towers Results Casson towers Any knot bounds a kinky disk in B 4 , i.e. a disk with transverse self-intersections. Any knot which bounds such a kinky disk with only positive self-intersections lies in P 0 . A Casson tower is built using layers of kinky disks, so they are natural objects to study in this context.
Introduction Goal Casson towers Results Casson towers K
Introduction Goal Casson towers Results Casson towers K
Introduction Goal Casson towers Results Casson towers K
Introduction Goal Casson towers Results Casson towers A Casson tower of height n consists of n layers of kinky disks. K
Introduction Goal Casson towers Results Casson towers A Casson tower of height n consists of n layers of kinky disks. A Casson tower T is of height (2 , n ) if it has two layers of kinky disks, and each member of a standard set of generators of π 1 ( T ) is in π 1 ( B 4 − T ) ( n ) . K
Introduction Goal Casson towers Results Casson towers Definition (R.) • A knot is in C n if it bounds a Casson tower of height n in B 4
Introduction Goal Casson towers Results Casson towers Definition (R.) • A knot is in C n if it bounds a Casson tower of height n in B 4 • A knot is in C + n if it bounds a Casson tower of height n in B 4 such that all the kinks at the initial disk are positive n if it bounds a Casson tower of height n in B 4 • A knot is in C − such that all the kinks at the initial disk are negative
Introduction Goal Casson towers Results Casson towers Definition (R.) • A knot is in C 2 , n if it bounds a Casson tower of height (2 , n ) in B 4
Introduction Goal Casson towers Results Casson towers Definition (R.) • A knot is in C 2 , n if it bounds a Casson tower of height (2 , n ) in B 4 • A knot is in C + 2 , n if it bounds a Casson tower of height (2 , n ) in B 4 such that all the kinks at the initial disk are positive • A knot is in C − 2 , n if it bounds a Casson tower of height (2 , n ) in B 4 such that all the kinks at the initial disk are negative
Introduction Goal Casson towers Results Results Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0 , G n +2 ⊆ F n
Introduction Goal Casson towers Results Results Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0 , G n +2 ⊆ F n Theorem (R.) For all n ≥ 0 , • C + n +2 ⊆ P n • C − n +2 ⊆ N n
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