Non-surjective satellite operators and piecewise-linear concordance Princeton University ICM Satellite Conference on Knots and Low Dimensional Manifolds August 22, 2014 Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Which knots K ⊂ R 3 (or S 3 ) can occur as cross-sections of embedded spheres in R 4 (or S 4 )? Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Which knots K ⊂ R 3 (or S 3 ) can occur as cross-sections of embedded spheres in R 4 (or S 4 )? Equivalently, which knots in R 3 (or S 3 ) bound properly embedded disks in R 4 + (or D 4 )? Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Definition A knot K ⊂ S 3 is (smoothly) slice if it is the boundary of a smoothly embedded disk in D 4 ; Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Definition A knot K ⊂ S 3 is (smoothly) slice if it is the boundary of a smoothly embedded disk in D 4 ; topologically slice if it is the boundary of a locally flat disk in D 4 (i.e., a continuously embedded disk with a normal bundle). Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Definition A knot K ⊂ S 3 is (smoothly) slice if it is the boundary of a smoothly embedded disk in D 4 ; topologically slice if it is the boundary of a locally flat disk in D 4 (i.e., a continuously embedded disk with a normal bundle). Knots K 1 , K 2 are smoothly/topologically concordant if they cobound an embedded annulus in S 3 × I , or equivalently if K 1 # − K 2 is topologically/smoothly slice, where − K = K r . Adam Simon Levine Non-surjective satellite operators and PL concordance
Concordance Definition A knot K ⊂ S 3 is (smoothly) slice if it is the boundary of a smoothly embedded disk in D 4 ; topologically slice if it is the boundary of a locally flat disk in D 4 (i.e., a continuously embedded disk with a normal bundle). Knots K 1 , K 2 are smoothly/topologically concordant if they cobound an embedded annulus in S 3 × I , or equivalently if K 1 # − K 2 is topologically/smoothly slice, where − K = K r . C top = { knots } / top. conc. C = { knots } / smooth conc. Adam Simon Levine Non-surjective satellite operators and PL concordance
Piecewise-linear concordance Since D 4 is the cone on S 3 , every knot bounds a piecewise-linear embedded disk in D 4 ! Adam Simon Levine Non-surjective satellite operators and PL concordance
Piecewise-linear concordance Since D 4 is the cone on S 3 , every knot bounds a piecewise-linear embedded disk in D 4 ! In other words, Dehn’s Lemma holds for D 4 . Adam Simon Levine Non-surjective satellite operators and PL concordance
Piecewise-linear concordance Since D 4 is the cone on S 3 , every knot bounds a piecewise-linear embedded disk in D 4 ! In other words, Dehn’s Lemma holds for D 4 . Conjecture (Zeeman, 1963) In an arbitrary compact, contractible 4 -manifold X other than the 4 -ball, not every knot K ⊂ ∂ X bounds a PL disk. Adam Simon Levine Non-surjective satellite operators and PL concordance
Piecewise-linear concordance Theorem (Matsumoto–Venema, 1979) There exists a non-compact, contractible 4 -manifold with boundary S 1 × R 2 such that S 1 × { pt } does not bound an embedded PL disk. Adam Simon Levine Non-surjective satellite operators and PL concordance
Piecewise-linear concordance Theorem (Matsumoto–Venema, 1979) There exists a non-compact, contractible 4 -manifold with boundary S 1 × R 2 such that S 1 × { pt } does not bound an embedded PL disk. Theorem (Akbulut, 1990) There exist a compact, contractible 4 -manifold X and a knot γ ⊂ ∂ X that does not bound an embedded PL disk in X. Adam Simon Levine Non-surjective satellite operators and PL concordance
Akbulut’s example Akbulut’s manifold X is the X original Mazur manifold: 0 X = S 1 × D 3 ∪ Q 2-handle , Q ⊂ S 1 × D 2 ⊂ ∂ ( S 1 × D 3 ) , γ = S 1 × { pt } . • γ Adam Simon Levine Non-surjective satellite operators and PL concordance
Akbulut’s example Akbulut’s manifold X is the X ′ original Mazur manifold: • X = S 1 × D 3 ∪ Q 2-handle , Q ⊂ S 1 × D 2 ⊂ ∂ ( S 1 × D 3 ) , γ = S 1 × { pt } . 0 But γ bounds a smoothly embedded disk in a different contractible 4-manifold X ′ with ∂ X ′ = ∂ X . γ Adam Simon Levine Non-surjective satellite operators and PL concordance
Akbulut’s example Akbulut’s manifold X is the X ′ original Mazur manifold: • X = S 1 × D 3 ∪ Q 2-handle , Q ⊂ S 1 × D 2 ⊂ ∂ ( S 1 × D 3 ) , γ = S 1 × { pt } . 0 But γ bounds a smoothly embedded disk in a different contractible 4-manifold X ′ with ∂ X ′ = ∂ X . γ In fact, X ′ ∼ = X , but not rel boundary. Adam Simon Levine Non-surjective satellite operators and PL concordance
Akbulut’s example Akbulut’s manifold X is the X original Mazur manifold: 0 X = S 1 × D 3 ∪ Q 2-handle , Q ⊂ S 1 × D 2 ⊂ ∂ ( S 1 × D 3 ) , γ = S 1 × { pt } . • But γ bounds a smoothly embedded disk in a different contractible 4-manifold X ′ with ∂ X ′ = ∂ X . γ In fact, X ′ ∼ = X , but not rel boundary. Adam Simon Levine Non-surjective satellite operators and PL concordance
Main theorem Theorem (L., 2014) X 0 There exist a contractible 4 -manifold X and a knot γ ⊂ ∂ X such that γ does not bound an embedded PL disk in any contractible manifold X ′ with • ∂ X ′ = ∂ X. γ Adam Simon Levine Non-surjective satellite operators and PL concordance
Main theorem Theorem (L., 2014) X 0 There exist a contractible 4 -manifold X and a knot γ ⊂ ∂ X such that γ does not bound an J embedded PL disk in any contractible manifold X ′ with • ∂ X ′ = ∂ X. In place of the trefoil, can use any knot J with ǫ ( J ) = 1, γ where ǫ is Hom’s concordance invariant. Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Signature: if K is slice, σ ( K ) = 0 (Murasugi) Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Signature: if K is slice, σ ( K ) = 0 (Murasugi) Tristram–Levine signatures Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Signature: if K is slice, σ ( K ) = 0 (Murasugi) Tristram–Levine signatures Algebraic concordance group (J. Levine) Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Signature: if K is slice, σ ( K ) = 0 (Murasugi) Tristram–Levine signatures Algebraic concordance group (J. Levine) Casson–Gordon invariants Adam Simon Levine Non-surjective satellite operators and PL concordance
Classical concordance obstructions There are many obstructions to a knot K ⊂ S 3 being topologically slice: Alexander polynomial: if K ⊂ S 3 is slice, ∆ K ( t ) = f ( t ) f ( t − 1 ) (Fox–Milnor) Signature: if K is slice, σ ( K ) = 0 (Murasugi) Tristram–Levine signatures Algebraic concordance group (J. Levine) Casson–Gordon invariants Freedman: If ∆ K ( t ) ≡ 1, then K is topologically slice; e.g., Whitehead doubles. But many such knots are not smoothly slice. Adam Simon Levine Non-surjective satellite operators and PL concordance
Smooth concordance obstructions For K ⊂ S 3 , we obtain several concordance invariants from knot Floer homology: Adam Simon Levine Non-surjective satellite operators and PL concordance
Smooth concordance obstructions For K ⊂ S 3 , we obtain several concordance invariants from knot Floer homology: τ ( K ) ∈ Z (Ozsváth–Szabó, Rasmussen): Adam Simon Levine Non-surjective satellite operators and PL concordance
Smooth concordance obstructions For K ⊂ S 3 , we obtain several concordance invariants from knot Floer homology: τ ( K ) ∈ Z (Ozsváth–Szabó, Rasmussen): τ ( K 1 # K 2 ) = τ ( K 1 ) + τ ( K 2 ) Adam Simon Levine Non-surjective satellite operators and PL concordance
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