Min-max Theory, Willmore conjecture, and Energy of links

Min-max Theory, Willmore conjecture, and Energy of links André Neves (Joint with Fernando Marques)

Q: What is the best way of immersing a sphere in space? A: The one that minimizes the bending energy H 2 = Σ Σ ( ) 2 k1 + k 2. 2

Q: What is the best way of immersing a sphere in space? A: The one that minimizes the bending energy H 2 = Σ Σ ( ) 2 k1 + k 2. 2 bending energy is conformally invariant; every compact surface has Σ H 2 4π with equality only for round sphere.

What is the best way of immersing a torus in space?

What is the best way of immersing a torus in space?

What is the best way of immersing a torus in space? Conjecture (Willmore, 65) For every torus Σ immersed in R 3 Σ H 2 2π 2.

Consider stereographic projection π : S 3 {north pole} R 3. If Σ S 3 then (1 + H 2 ) = H 2. Σ π(σ) the Willmore energy of Σ S 3 is defined to be W(Σ) = (1 + H 2 ). Σ

Consider stereographic projection π : S 3 {north pole} R 3. If Σ S 3 then (1 + H 2 ) = H 2. Σ π(σ) the Willmore energy of Σ S 3 is defined to be W(Σ) = (1 + H 2 ). Willmore Conjecture - 65 Every torus Σ immersed in S 3 has W(Σ) 2π 2. Σ

Theorem (Marques-N., 12) For every embedded surface Σ S 3 with genus g 1 W(Σ) 2π 2 with equality if and only if Σ is conformal to S 1 ( 1 2 ) S 1 ( 1 2 ). Corollary A Willmore conjecture holds. Corollary B The only two minimal surfaces in S 3 with area 2π 2 are great spheres and Clifford torus.

Links in space γ 1 and γ 2 two linked curves in R 3.

Links in space γ 1 and γ 2 two linked curves in R 3. lk (γ 1, γ 2 ) is the linking number lk = 1 lk = 3

Links in space Möbius cross energy of a link (γ 1, γ 2 ) is γ 1 E(γ 1, γ 2 ) = (s) γ 2 (t) ds dt. γ 1 (s) γ 2 (t) 2 S 1 S 1 E(γ 1, γ 2 ) is conformally invariant and E(γ 1, γ 2 ) 4π lk (γ 1, γ 2 )

Links in space Q: What is the best way of immersing a non-trivial link (γ 1, γ 2 ) in R 3? A: The one that minimizes the Möbius energy.

Links in space Q: What is the best way of immersing a non-trivial link (γ 1, γ 2 ) in R 3? A: The one that minimizes the Möbius energy. Conjecture (Freedman-He-Wang, 94) If lk (γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. If true, every non-trivial link has E(γ 1, γ 2 ) 2π 2.

Theorem (Agol-Marques-N., 12) If lk (γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2.

Theorem (Agol-Marques-N., 12) If lk (γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. If equality then (γ 1, γ 2 ) is conformal to the standard Hopf link in S 3 β 1 (t) = (cos t, sin t, 0, 0) and β 2 (s) = (0, 0, cos s, sin s).

Partial results... (Willmore 71, Shioama-Takagi 70): Conjecture is true for tubes of constant radius around a space curve.

Partial results... (Willmore 71, Shioama-Takagi 70): Conjecture is true for tubes of constant radius around a space curve. (Langevin-Rosenberg 76): If Σ R 3 is a knotted torus then W(Σ) 8π.

Partial results... (Willmore 71, Shioama-Takagi 70): Conjecture is true for tubes of constant radius around a space curve. (Langevin-Rosenberg 76): If Σ R 3 is a knotted torus then W(Σ) 8π. (Chen, 83): Conjecture is true for flat tori in 3-sphere.

Partial results... (Willmore 71, Shioama-Takagi 70): Conjecture is true for tubes of constant radius around a space curve. (Langevin-Rosenberg 76): If Σ R 3 is a knotted torus then W(Σ) 8π. (Chen, 83): Conjecture is true for flat tori in 3-sphere. (Langer-Singer 84): Conjecture is true for tori of revolution.

Partial results... (Li-Yau 82): If Σ S 3 covers a point k times then W(Σ) 4πk. Thus Σ not embedded = W(Σ) 8π.

Partial results... (Li-Yau 82): If Σ S 3 covers a point k times then W(Σ) 4πk. Thus Σ not embedded = W(Σ) 8π. (Li-Yau 82): Conjecture is true for a set of conformal classes which contains the conformal class of the Clifford torus.

Partial results... (Li-Yau 82): If Σ S 3 covers a point k times then W(Σ) 4πk. Thus Σ not embedded = W(Σ) 8π. (Li-Yau 82): Conjecture is true for a set of conformal classes which contains the conformal class of the Clifford torus. (Montiel-Ros 86):Conjecture is true for a larger set of conformal classes than the set given by Li-Yau.

Partial results... (Benisom-Mutz 91) Conjecture is true for observed toroidal vesicles in some cells.

Partial results... (Simon 93): There is a torus which has least Willmore energy among all tori.

Partial results... (Simon 93): There is a torus which has least Willmore energy among all tori. (Ros 99, Topping 00): Conjecture is true for tori symmetric under the antipodal map.

Partial results... (Simon 93): There is a torus which has least Willmore energy among all tori. (Ros 99, Topping 00): Conjecture is true for tori symmetric under the antipodal map. (Ros 00): Conjecture is true for tori symmetric with respect to a point.

Min-max Theory Z 2 (S 3 ) = { integral 2-currents with no boundary in S 3 }. M : Z 2 (S 3 ) R, M(Σ) = mass of Σ.

Min-max Theory Z 2 (S 3 ) = { integral 2-currents with no boundary in S 3 }. M : Z 2 (S 3 ) R, M(Σ) = mass of Σ. φ : I n Z 2 (S 3 ) continuous in flat norm, where I n = k-cube.

Min-max Theory Z 2 (S 3 ) = { integral 2-currents with no boundary in S 3 }. M : Z 2 (S 3 ) R, M(Σ) = mass of Σ. φ : I n Z 2 (S 3 ) continuous in flat norm, where I n = k-cube. [φ] = {ψ homotopic to φ relative to the boundary (i.e. ψ I n = φ I n).}

Min-max Theory Z 2 (S 3 ) = { integral 2-currents with no boundary in S 3 }. M : Z 2 (S 3 ) R, M(Σ) = mass of Σ. φ : I n Z 2 (S 3 ) continuous in flat norm, where I n = k-cube. [φ] = {ψ homotopic to φ relative to the boundary (i.e. ψ I n = φ I n).} L([φ]) = inf ψ [φ] sup x I n M(ψ(x)).

Min-max Theory Theorem (Pitts, 81) Assume L([φ]) = (this means [φ] 0 π k (Z 2 (S 3 ), φ I k ).) inf sup M(ψ(x)) > sup M(φ(x)), ψ [φ] x I k x I k There is Σ S 3 smooth embedded minimal surface (with multiplicities) so that L([φ]) = M(Σ).

Min-max Theory Theorem (Pitts, 81) Assume L([φ]) = (this means [φ] 0 π k (Z 2 (S 3 ), φ I k ).) inf sup M(ψ(x)) > sup M(φ(x)), ψ [φ] x I k x I k There is Σ S 3 smooth embedded minimal surface (with multiplicities) so that Conjecture: index(σ) k. L([φ]) = M(Σ).

Min-max Theory Theorem (Pitts, 81) Assume L([φ]) = (this means [φ] 0 π k (Z 2 (S 3 ), φ I k ).) inf sup M(ψ(x)) > sup M(φ(x)), ψ [φ] x I k x I k There is Σ S 3 smooth embedded minimal surface (with multiplicities) so that L([φ]) = M(Σ). Conjecture: index(σ) k. Theorem (Almgren, 62) π k (Z 2 (S 3 ), 0) = 0 if k 1; π 1 (Z 2 (S 3 ), 0) = Z

Min-max Theory Theorem (Pitts, 81) Assume L([φ]) = (this means [φ] 0 π k (Z 2 (S 3 ), φ I k ).) inf sup M(ψ(x)) > sup M(φ(x)), ψ [φ] x I k x I k There is Σ S 3 smooth embedded minimal surface (with multiplicities) so that L([φ]) = M(Σ). Conjecture: index(σ) k. Theorem (Almgren, 62) π k (Z 2 (S 3 ), 0) = 0 if k 1; π 1 (Z 2 (S 3 ), 0) = Z and given Σ = A surface then φ 0 : [ π, π] Z 2 (S 3 ), φ 0 (t) = {x : dist(x, Σ) < t} has π 1 (Z 2 (S 3 ), 0) = Z[φ 0 ].

Min-max Theory Theorem (Urbano, 90) Σ S 3 compact embedded minimal surface with index(σ) 5. Then either Σ is a great sphere (index(σ) = 1 and area(σ) = 4π), or Σ is the Clifford torus (index(σ) = 5 and area(σ) = 2π 2 ).

Min-max Theory Theorem (Urbano, 90) Σ S 3 compact embedded minimal surface with index(σ) 5. Then either Σ is a great sphere (index(σ) = 1 and area(σ) = 4π), or Σ is the Clifford torus (index(σ) = 5 and area(σ) = 2π 2 ). L([φ 0 ]) = 4π where φ 0 = Almgren s sweepout. How to obtain φ with L([φ]) = 2π 2?

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π]

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π] Theorem (Marques-N.) Consider φ : I 5 Z 2 (S 3 ) continuous with

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π] Theorem (Marques-N.) Consider φ : I 5 Z 2 (S 3 ) continuous with 1 φ(i 4 {0}) = φ(i 4 {1}) = 0;

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π] Theorem (Marques-N.) Consider φ : I 5 Z 2 (S 3 ) continuous with 1 φ(i 4 {0}) = φ(i 4 {1}) = 0; 2 φ( I 4 I) R and for x I 4, φ(x, t) T t = 1/2;

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π] Theorem (Marques-N.) Consider φ : I 5 Z 2 (S 3 ) continuous with 1 φ(i 4 {0}) = φ(i 4 {1}) = 0; 2 φ( I 4 I) R and for x I 4, φ(x, t) T t = 1/2; 3 the sweepout t φ(1/2, 1/2, 1/2, 1/2, 1/2, t) is non-trivial.

Min-max Theory T = {oriented great spheres} S 3 R = {oriented round spheres} S 3 [0, π] Theorem (Marques-N.) Consider φ : I 5 Z 2 (S 3 ) continuous with 1 φ(i 4 {0}) = φ(i 4 {1}) = 0; 2 φ( I 4 I) R and for x I 4, φ(x, t) T t = 1/2; 3 the sweepout t φ(1/2, 1/2, 1/2, 1/2, 1/2, t) is non-trivial. If φ : I 4 {1/2} S 3 T S 3 has non-zero degree then L([φ]) > 4π = max x I 5 M(φ(x)).

Min-max Theory Corollary For φ as in previous theorem, sup x I 5 M(φ(x)) 2π 2.

Min-max Theory Corollary For φ as in previous theorem, sup x I 5 M(φ(x)) 2π 2. Idea: (Previous Thm) = L([φ]) > sup x I 5 M(φ(x)) = 4π

Min-max Theory Corollary For φ as in previous theorem, sup x I 5 M(φ(x)) 2π 2. Idea: (Previous Thm) = L([φ]) > sup x I 5 M(φ(x)) = 4π (Pitts Thm) = Σ minimal: L([φ]) = M(Σ) > 4π and index(σ) 5

Min-max Theory Corollary For φ as in previous theorem, sup x I 5 M(φ(x)) 2π 2. Idea: (Previous Thm) = L([φ]) > sup x I 5 M(φ(x)) = 4π (Pitts Thm) = Σ minimal: L([φ]) = M(Σ) > 4π and index(σ) 5 (Urbano s Thm) = Σ = Clifford torus = sup x I 5 M(φ(x)) L([φ]) = 2π 2.

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ].

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ]. Set K = ψ 1 (T ), where K I 4 {1/2}.

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ]. Set K = ψ 1 (T ), where K I 4 {1/2}. (A) [ K ] = [ I 4 {1/2}] in H 3 ( I 4 {1/2}, Z)

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ]. Set K = ψ 1 (T ), where K I 4 {1/2}. (A) [ K ] = [ I 4 {1/2}] in H 3 ( I 4 {1/2}, Z) φ = ψ on K and ψ : K T S 3 = degree(φ I 4 {1/2}) = degree(ψ K ) = 0

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ]. Set K = ψ 1 (T ), where K I 4 {1/2}. (A) [ K ] = [ I 4 {1/2}] in H 3 ( I 4 {1/2}, Z) φ = ψ on K and ψ : K T S 3 = degree(φ I 4 {1/2}) = degree(ψ K ) = 0 (B) [ K ] = 0 in H 3 ( I 4 {1/2}, Z)

Sketch of proof of Theorem: Assume L([φ]) = 4π = sup x I k M (ψ(x)) for some ψ [φ]. Set K = ψ 1 (T ), where K I 4 {1/2}. (A) [ K ] = [ I 4 {1/2}] in H 3 ( I 4 {1/2}, Z) φ = ψ on K and ψ : K T S 3 = degree(φ I 4 {1/2}) = degree(ψ K ) = 0 (B) [ K ] = 0 in H 3 ( I 4 {1/2}, Z) ψ c non-trivial sweepout with sup M(ψ c(t)) sup M(ψ(x)) = 4π t I x I k = ψ c(i) T = c(i) K.

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2.

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v 2 C : B 4 [ π, π] Z 2 (S 3 ), C(v, t) = {x : dist(x, F v (Σ)) < t}

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v 2 C : B 4 [ π, π] Z 2 (S 3 ), C(v, t) = {x : dist(x, F v (Σ)) < t} 3 (Ros, 99) M(C(v, t)) W(C(v, 0)) = W(F v (Σ)) = W(Σ)

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v 2 C : B 4 [ π, π] Z 2 (S 3 ), C(v, t) = {x : dist(x, F v (Σ)) < t} 3 (Ros, 99) M(C(v, t)) W(C(v, 0)) = W(F v (Σ)) = W(Σ) 4 C S 3 [ π,π] is not continuous v i B 4 p / Σ = F vi (Σ) { p} v i B 4 p Σ with slope θ = F vi (Σ) B π 2 θ ( cos θn(p) sin θp).

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v 2 C : B 4 [ π, π] Z 2 (S 3 ), C(v, t) = {x : dist(x, F v (Σ)) < t} 3 (Ros, 99) M(C(v, t)) W(C(v, 0)) = W(F v (Σ)) = W(Σ) 4 C S 3 [ π,π] is not continuous v i B 4 p / Σ = F vi (Σ) { p} v i B 4 p Σ with slope θ = F vi (Σ) B π 2 θ ( cos θn(p) sin θp). 5 Reparametrize C to obtain φ : B 4 [ π, π] Z 2 (S 3 ) continuous with φ(s 3 [ π, π]) R and φ(x, t) T t = 0; 6 degree(φ S3 {0}) = genus of Σ!

Theorem (Marques-N.) If Σ S 3 has positive genus then W(Σ) 2π 2. 1 Given v B 4 consider F v Conf (S 3 ), x 1 v 2 x v 2 (x v) v 2 C : B 4 [ π, π] Z 2 (S 3 ), C(v, t) = {x : dist(x, F v (Σ)) < t} 3 (Ros, 99) M(C(v, t)) W(C(v, 0)) = W(F v (Σ)) = W(Σ) 4 C S 3 [ π,π] is not continuous v i B 4 p / Σ = F vi (Σ) { p} v i B 4 p Σ with slope θ = F vi (Σ) B π 2 θ ( cos θn(p) sin θp). 5 Reparametrize C to obtain φ : B 4 [ π, π] Z 2 (S 3 ) continuous with φ(s 3 [ π, π]) R and φ(x, t) T t = 0; 6 degree(φ S3 {0}) = genus of Σ! 7 Corollary to our Min-max Thm can be applied and so 2π 2 sup (v,t) B 4 [ π,π] M(φ(x)) = sup M(C(v, t)) W(Σ) (v,t) B 4 [ π,π]

Min-max Theory Continuous I n = n-cube; Discrete I(n, k) vertices of 3 nk subdivisions of I n ;

Min-max Theory Continuous I n = n-cube; φ : I n Z 2 (S 3 ) continuous in the flat topology; Discrete I(n, k) vertices of 3 nk subdivisions of I n ; sequence S = {φ i } i N with φ i : I(n, k i ) Z 2 (S 3 ), where x, y adjacent vertices = M(φ i (x) φ i (y)) δ i 0;

Min-max Theory Continuous I n = n-cube; φ : I n Z 2 (S 3 ) continuous in the flat topology; sup{m(φ(x)) : x I n }; Discrete I(n, k) vertices of 3 nk subdivisions of I n ; sequence S = {φ i } i N with φ i : I(n, k i ) Z 2 (S 3 ), where x, y adjacent vertices = M(φ i (x) φ i (y)) δ i 0; L(S) = lim i sup{m(φ i (x)) : x I(n, k i )};

Min-max Theory Continuous I n = n-cube; φ : I n Z 2 (S 3 ) continuous in the flat topology; sup{m(φ(x)) : x I n }; define [φ] π n (Z 2 (S 3 ), φ I n); Discrete I(n, k) vertices of 3 nk subdivisions of I n ; sequence S = {φ i } i N with φ i : I(n, k i ) Z 2 (S 3 ), where x, y adjacent vertices = M(φ i (x) φ i (y)) δ i 0; L(S) = lim i sup{m(φ i (x)) : x I(n, k i )}; define Π = [S] π n # (Z 2 (S 3 ), φ I n) (need φ I n continuous in varifold norm);

Min-max Theory Continuous I n = n-cube; φ : I n Z 2 (S 3 ) continuous in the flat topology; sup{m(φ(x)) : x I n }; define [φ] π n (Z 2 (S 3 ), φ I n); L([φ]) = inf ψ [φ] sup{m(ψ(x)) : x I n }. Discrete I(n, k) vertices of 3 nk subdivisions of I n ; sequence S = {φ i } i N with φ i : I(n, k i ) Z 2 (S 3 ), where x, y adjacent vertices = M(φ i (x) φ i (y)) δ i 0; L(S) = lim i sup{m(φ i (x)) : x I(n, k i )}; define Π = [S] π n # (Z 2 (S 3 ), φ I n) (need φ I n continuous in varifold norm); L([S]) = inf S [S] L(S).

Min-Max Theory Pitts Theorem Consider [S] π # n (Z 2 (S 3 ), φ I n) with L([S]) > sup{m(φ(x)) : x I n }.

Min-Max Theory Pitts Theorem Consider [S] π # n (Z 2 (S 3 ), φ I n) with L([S]) > sup{m(φ(x)) : x I n }. There is Σ a smooth embedded minimal surface (with multiplicities) so that L([S]) = inf L(S) = M(Σ). S [S]

Min-Max Theory Pitts Theorem Consider [S] π # n (Z 2 (S 3 ), φ I n) with L([S]) > sup{m(φ(x)) : x I n }. There is Σ a smooth embedded minimal surface (with multiplicities) so that L([S]) = inf L(S) = M(Σ). S [S] Interpolation Theorem Given φ : I n Z 2 (S 3 ) continuous in the flat topology, there exists k i N and φ i : I(n, k i ) 0 Z 2 (S 3 ) so that: (a) S = [{φ i } i N ] π # n (Z 2 (S 3 ), φ I n),

Min-Max Theory Pitts Theorem Consider [S] π # n (Z 2 (S 3 ), φ I n) with L([S]) > sup{m(φ(x)) : x I n }. There is Σ a smooth embedded minimal surface (with multiplicities) so that L([S]) = inf L(S) = M(Σ). S [S] Interpolation Theorem Given φ : I n Z 2 (S 3 ) continuous in the flat topology, there exists k i N and φ i : I(n, k i ) 0 Z 2 (S 3 ) so that: (a) S = [{φ i } i N ] π # n (Z 2 (S 3 ), φ I n), (b) L({φ i }) sup{m(φ(x)) : x I n },

Min-Max Theory Pitts Theorem Consider [S] π # n (Z 2 (S 3 ), φ I n) with L([S]) > sup{m(φ(x)) : x I n }. There is Σ a smooth embedded minimal surface (with multiplicities) so that L([S]) = inf L(S) = M(Σ). S [S] Interpolation Theorem Given φ : I n Z 2 (S 3 ) continuous in the flat topology, there exists k i N and φ i : I(n, k i ) 0 Z 2 (S 3 ) so that: (a) S = [{φ i } i N ] π # n (Z 2 (S 3 ), φ I n), (b) L({φ i }) sup{m(φ(x)) : x I n }, (c) Moreover, L([S]) = L([φ]). lim sup{f(φ i(x) φ(x)) x I(n, k i ) 0 } = 0. i

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2.

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t).

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t).

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t). area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). Given v B 4, F v (x) = Conf (R 4 ). F v (B 1 (0)) = B r(v) (c(v)). x v x v 2

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t). area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). Given v B 4, F v (x) = Conf (R 4 ). F v (B 1 (0)) = B r(v) (c(v)). x v x v 2 1 φ : B 4 [0, + ] Z 2 (S 3 ), φ(v, λ) = G(F v (γ 1 ), λ(f v (γ 2 ) c(v)) + c(v)).

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t). area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). Given v B 4, F v (x) = Conf (R 4 ). F v (B 1 (0)) = B r(v) (c(v)). x v x v 2 1 φ : B 4 [0, + ] Z 2 (S 3 ), φ(v, λ) = G(F v (γ 1 ), λ(f v (γ 2 ) c(v)) + c(v)). 2 area(φ(v, λ)) E(φ(v, 1)) = E(F v (γ 1 ), F v (γ 2 )) = E(γ 1, γ 2 ).

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t). area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). Given v B 4, F v (x) = Conf (R 4 ). F v (B 1 (0)) = B r(v) (c(v)). x v x v 2 1 φ : B 4 [0, + ] Z 2 (S 3 ), φ(v, λ) = G(F v (γ 1 ), λ(f v (γ 2 ) c(v)) + c(v)). 2 area(φ(v, λ)) E(φ(v, 1)) = E(F v (γ 1 ), F v (γ 2 )) = E(γ 1, γ 2 ). 3 Given v S 3, φ(v, 1) = lk(γ 1, γ 2 ) B π/2 ( v) = degree(φ S3 {1}) = lk(γ 1, γ 2 ) 0

Theorem (Agol-Marques-N.) If (γ 1, γ 2 ) link in S 3 has lk(γ 1, γ 2 ) = ±1, then E(γ 1, γ 2 ) 2π 2. Given (σ 1, σ 2 ) R 4 consider G(σ 1, σ 2 ) : S 1 S 1 S 3 G(σ 1, σ 2 )(s, t) = (σ 1(s) σ 2 (t)) σ 1 (s) σ 2 (t). area(g(σ 1, σ 2 )) E(γ 1, γ 2 ). Given v B 4, F v (x) = Conf (R 4 ). F v (B 1 (0)) = B r(v) (c(v)). x v x v 2 1 φ : B 4 [0, + ] Z 2 (S 3 ), φ(v, λ) = G(F v (γ 1 ), λ(f v (γ 2 ) c(v)) + c(v)). 2 area(φ(v, λ)) E(φ(v, 1)) = E(F v (γ 1 ), F v (γ 2 )) = E(γ 1, γ 2 ). 3 Given v S 3, φ(v, 1) = lk(γ 1, γ 2 ) B π/2 ( v) = degree(φ S3 {1}) = lk(γ 1, γ 2 ) 0 4 Can apply our Min-max Thm to conclude 2π 2 max (v,t) B 4 [0,+ ] area(φ(v, t)) E(γ 1, γ 2 ).