Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo (indeterminacy ).,,.,, (I), Ω m., ( 6.2)., ( 6.3).,,,., ()Milnor,.,. 2,.. G, Γ k G,, Γ 1 G := G, Γ 2 G := [G, G], Γ 3 G := [[G, G], G],..., Γ m G := [Γ m 1 G, G],. F q, x 1,..., x q., Q m Γ m 1 F/Γ m F. 0 Q m F/Γ m F p m 1 F/Γm 1 F 0 (central extension). (1) Q m,, (Hall basis) ([CFL, Theorem 1.5] ).,. 1, JSPS (:25800049). E-mail address: nosaka@math.kyushu-u.ac.jp

L S 3, q-. - (m l, l l ), preferred ( l q). f 2 : π 1 (S 3 \ L) F/Γ 2 F = Z q., [M1, IO],. m N, : A m. k m, f k : π 1 (S 3 \ L) F/Γ k F π 1 (S 3 \ L) f m f 2 f 3 f 4 F/Γ 2 F F/Γ 3 F F/Γ 4 F F/Γ m F. p 2 p 3 p m 1., f m, f m f m (, )., [x i ] F/Γ m F, {x k i } k Z Q m 2., f m preferred l l Q m ( f m (m l ) ±1 )., L, q-tuple : ( fm (l 1 ),..., f m (l q ) ) (Q m ) q., A m+1., 2.1 ([M2]). A m., f m f m+1 : π 1 (S 3 \ L) F/Γ m+1 F, m,, l q, f m (l l ) = 0 Q m.,,.,. : (I) F/Γ m F., ( ) Z X 1,..., X q. f m (l l ) ; [M2, IO]. (II), l l. Milnor [M2, 3] ([Hil, Chapters 1 & 11 14] ), f m (l l ) l l ( (7))., m q = #L. (III), f m, F/Γ m F. (IV), Milnor [M2], I {1, 2,..., q} m.,. (V), Milnor, ([M2, Hil, St] ).,. 2,, m.

3 Magnus., Gupta-Gupta[GG] Magnus.,, F/Γ m F.. Ω m Z[λ (j) i ],, λ (j) i i {1, 2,..., m 1}, j {1,..., q} (, qm q)., Υ m : F GL m (Ω m ) : Υ m (x j ) = 1 λ (j) 1 0 0 0 1 λ (j) 2 0......... 0 0 1 λ (j) m 1 0 0 0 1, (cf.,, ). I m. (i) y k Γ k F, Υ m (y) I m, (1, 2)-, (1, 3)-,..., (1, k 1)- ( k < m). (ii), m Γ m F, Υ m (Γ m F ) = {I m }. (iii) [GG], F/Γ m F GL m (Ω m ). (iv) Υ m (Q m ), (1, m)- (, )., (2). E i,j, (i, j)-., b Im(Υ m ), B p 1 m (b) Im(Υ m+1 )., (iv), B, (1, m + 1)-.,, Im(Υ m+1 ) :. B 1 AB = (B + ωe 1,m+1 ) 1 A(B + ωe 1,m+1 ), A Im(Υ m+1 ), ω Ω m+1. (2) 4. 4.1 4.2. (II)(III).. L D, A m. 1, α 1, α 2,..., α Nj l j. α 1 m j. β k α k α k+1, ϵ k {±1} α k β k. Wirtinger, f m { arc of D } F/Γ m F., k, C : {α k } k Nj Im(Υ m+1 ) : C(α 1 ) = Υ m+1 (x j ) (., 1 < k N j, B k p 1 m fm (β k ) ), C(α k+1 ) B ϵ k k C(α k ) B ϵ k k. (2), B k., C : { arc of D } Im(Υ m+1 ).

l j β1 β 2 β Nj α 1 α 2 α 3 1: D, l j, α i, β i. 4.1, N j -. : Ψ m (j) := Υ m+1 (x j ) 1 B ϵ N j N j C(α Nj ) B ϵ N j N j Υ m+1 (Q m+1 ). (3), Ψ m (j) l j ( j q = #L)., j #L, I j : Γ m F Γ m+1 F ; y x 1 j y 1 x j y, (4). I j : Υ m (Q m ) Υ m+1 (Q m+1 )., I j, Ψ m (j) Milnor : 4.1. I j, I j ( Υm f m (l j ) ) = Ψ m (j).. 5.1 (5).. B ϵ 1 1 B ϵ 2 2 B ϵ N j N j B, [g, h] = ghg 1 h 1.,, Ψ m (j) = Υ m ([x 1 j, B 1 ]). [x j, z] = [x j, p m (z)] Q m+1 ( z Γ m 1 F ). 1 p m (B) = f m (l j ), I j ( Υm f m (l j ) ) = Υ m ([x 1 j, f m (l 1 j )]) = Υ m ([x 1 j, B 1 ]) = Ψ m (j).., 2.1, f m+1 : 4.2. m f m (l l ), l #L,., C : { arc of D } Im(Υ m+1 ) f m+1 : π 1 (S 3 \ L) Υ m+1 (F/Γ m+1 F ).. C, j 1 k < N j, k- Wirtinger. 4.1, Ψ m (j) = 0 N j Wirtinger., C, Wirtinger.,., 2.1.,. f 2 : π 1 (S 3 \ L) Z q., k h f k : π 1 (S 3 \ L) F/Γ k F (, A h ), h- Milnor f h (l l ). 4.2, f h+1., l f m (l l ) 0 m., 4.1, m- Milnor., Ω m, Mathematica., (I) (III),. 5,., c(l) L, lk(l) Z.

5.1,. (IV),, Ω m. r, s N, ι s : Ω r Ω r+s. κ r : Ω s Ω r+s κ r (λ (j) i ) = λ (j) i+r ( r ). : [, ] : Ω r κ r (Ω s ) Ω r+s ; (a, κ r (b)) ι s (a)κ r (b) ι r (b)κ s (a). 3 (iv), Υ s (Q s ) Ω s, [, ] : Υ r (Q r ) κ r ( Υs (Q s ) ) Υ r+s (Q r+s ).,,. Υ r+s (ghg 1 h 1 ) = [Υ r (g), κ r ( Υs (h) ) ], g Γ r 1 F, and h Γ s 1 F (5)., Υ r+s (ghg 1 h 1 ),. [CFL] Corollaries 2.2 2.3.,,, (Jacobi) (S1), (S2), (S2 ), (S3), Q m., (5), Milnor (Table 1 )... S 2., J = (j 1 j n ) {1,..., q} n σ = (σ 1,..., σ n 1 ) (S 2 ) n 1, σ(j) = (j σ 1,..., j σ n) N n : (j σ 1,..., j σ n) = σ n 1 (σ n 2 ( σ 4 (σ 3 (σ 2 (σ 1 (j 1, j 2 ), j 3 ), j 4 ), j 5 ) ), j n )., : [[ [[λ (j 1) 1, λ (j 2) 2 ], λ (j 3) 3 ] ], λ (jn) n ] = sign(σ) λ (jσ 1 ) 1 λ (jσ 2 ) 2 λ (jσ n) n Ω n+1. σ (S 2 ) n 1, (5) () : Υ m ([[ [[x j1, x j2 ], x j3 ] ], x jm 1 ]) = [[ [[λ (j 1) 1, λ (j 2) 2 ], λ (j 3) 3 ] ], λ (j m 1) m 1 ]. (6) α γ β x 1 x 2 x 1 x 2 x 3 x 4 x m 2 x m 1 x m 2: Whitehead Milnor.

5.2 Whitehead,., Whitehead (cf. [IO, 10.3], [Mu1, 8][St]). x 1, x 2 α, β, γ, 2. < 4. m = 4. Wirtinger, C : {arcs of D} GL 5 (Ω 5 ) C(α) = C(x 2 )C(x 1 )C(x 2 ) 1, C(β) = C(α) 1 C(x 1 )C(α), C(γ) = C(β) 1 C(α)C(β).., f(x i ) = Υ 5 (x i ), C(α) C(β). C(β) = C(α) = 1 λ (1) 1 [λ (2) 1, λ(1) 2 ] [λ(1) 1, λ(2) 2 ]λ(2) 3 [λ (1) 1, λ(2) 2 ]λ(2) 3 λ(2) 4 0 1 λ (1) 2 [λ (2) 2, λ(1) 3 ] [λ(1) 2, λ(2) 3 ]λ(2) 4 0 0 1 λ (1) 3 [λ (2) 3, λ(1) 4 ] 0 0 0 1 λ (1) 4 0 0 0 0 1 1 λ (1) 1 0 [[λ (1) 1, λ(2) 2 ], λ(1) 3 ] [[λ(1) 2, λ(2) 3 ], λ(1) 4 ]λ(1) 1 [[λ (1) 0 1 λ (1) 2 0 [[λ (1) 0 0 1 λ (1) 1, λ(2) 2 ], λ(1) 3 ]λ(2) 2, λ(2) 3 ], λ(1) 4 ] 3 0 0 0 0 1 λ (1) 4 0 0 0 0 1, 4 [λ (1) 1, λ(2) 2 ][λ(1) 3, λ(2) 4 ] (,, 4 4, C f 4 : π 1 (S 3 \L) GL 4 (Ω 4 ) )., (13) Ψ 4 (j), : Ψ 4 (1) = Υ 5 (x 1 ) 1 C(α)C(x 2 ) 1 C(β)C(x 2 )Υ 5 (x 1 )C(x 2 ) 1 C(β) 1 C(x 2 )C(α) 1, Ψ 4 (2) = Υ 5 (x 2 ) 1 C(x 1 )C(β) 1 Υ 5 (x 2 )C(β)C(x 1 ) Υ 5 (Q 5 ). (Mathematica ), (1, 5)-: Ψ 4,1 (N 1 ) (1,5) = Ψ 4,2 (N 2 ) (1,5) = [[[λ (1) 3 ], λ (2) 4 ] = λ (1) 3 λ (1) 4 λ (2) 1 λ (2) 2 2λ (1) 2 λ (1) 4 λ (2) 1 λ (2) 3 + 2λ (1) 1 λ (1) 3 λ (2) 2 λ (2) 4 λ (1) 1 λ (1) 2 λ (2) 3 λ (2) 4.,. PC, Milnor, cl(l), m < 11 cl(l) < 20., lk(l) = 0 c(l) < 9 2- L,., 1,, : Υ := [[[λ (1) 3 ], λ (2) 4 ] Ω 5, Λ := [[[[[λ (1) 4 ], λ (1) 5 ], λ (2) 6 ] Ω 7., lk(l) = 0 c(l) = 9.,, Υ Λ., lk(l) = 0 2, Milnor., L m, ( 2 ). k- l k π 1 (S 3 \ L m ), Ab(l k ) = 0. 5.1. L m Υ m f m (l k ) (1,m),. ( 1) m k+1[( [[ [[λ (1) 2 ], λ (3) 3 ] ], λ (k 1) k 1, ]]), ( [[ [[λ (m) k, λ (m 1) k+1 ], λ (m 2) k+2 ] ], λ m (k+1) ] )]. [M2, HM] k = 1, k.

Link 5 2 1 7 2 4 7 2 6 7 2 8 8 2 10 8 2 12 8 2 13 8 2 15 m 4 4 4 4 6 6 4 4 Ψ m (1) Υ 2Υ Υ Υ Λ Λ Υ Υ Ψ m (2) Υ 2Υ Υ Υ Λ Λ Υ Υ 1: 0, Milnor. 6 µ-. f m (l l ) 0., [M1, M2] Milnor. indeteminacy,., µ-,. (V),.. Milnor [M2] ([Hil, 12] )., h- π 1 (S 3 \ L)/Γ h π 1 (S 3 \ L) : x1,..., x q [xj, w j ] = 1 for j q, Γ h F, (7) x j w j, j- (w j h )., F/Γ h F. F/Γ h F = ImΥ h, N h,l, (7) ImΥ h /N h,l f h., 2.1, w j Γ m F, [x j, w j ] Γ m+1 F., w j, [x j, w j ] f h., [x j, w j ], ImΥ h /N h,l., h, N h,l f h : π 1 (S 3 \ L) Im(Υ h )/N h,l, f h 1 = [p h 1 ] f h f h (m j ) = [Υ h (x j )]., ImΥ h K h., A m h = m, N m,l = 0 f m = f m., f h N h,l K h. p h s h : Im(Υ h ) Im(Υ h+1 ). (3), 1 α k, β k. s h (K h ) Im(Υ h+1 ), s h (K h )., C h (α k ), Im(Υ h+1 )/ s h (K h ), :, C h (α 1 ) = Υ h+1 (x j ),, 1 < k N j, B k Im(Υ h+1 ) [p h (B k )] = f h (β k ), C h (α k+1 ) B ϵ k k C h (α k ) B ϵ k k., N j -, : µ h L(j) := Υ m+1 (x j ) 1 B ϵ N j N j C h (α Nj ) B ϵ N j N j Im(Υ h+1 ))/ s h (K h ). (8), µ h L (j) Υ h+1(q h+1 )/ ( s h (K h ) Υ h (Q h ) ) ( p h ( µ h L (j))) = µh 1 L (j) = 0 )., N h+1,l, s h (K h ) µ h L (1),..., µh L (#L)., N h+1,l = s h (K h ), µ h L(1),..., µ h L(#L).

, 4.2, C h f h+1 : π 1 (S 3 \ L) Im(Υ h+1 )/N h+1,l, : π 1 (S 3 \ L) f h+1 f m f m+1 f h Im(Υ m ) Im(Υ m+1 )/N m+1,l Im(Υ h )/N h,l Im(Υ h+1 )/N h+1,l. p m p h 1 4.1 modulo N h,l, : 6.1. j #L, µ h L (j) [Υ h+1, s h ( f h (l j ))]., 4.1., : 6.2. h- µ-, #L-tuple ( µ h L (1),..., µh L (#L))., µ-., I, Z (I)., ( ),., µ-, N h,l Ω m,,.,, µ-.,, ( s h, ). 6.3. f h, : π 1 (S 3 \ L)/Γ h π 1 (S 3 \ L) = ImΥ h /N h,l.. h = m, (7). h. f h+1 h π 1 (S 3 \ L)/Γ h π 1 (S 3 \ L)., 6.1 f h+1 ([x j, w j ]) = µ h L (j), f h+1., µ-, (7),,. p h 7 µ-, µ-., (8) N h,l Υ h+1 (Q h+1 ), ( 7.1). 5.1, Υ h (Q h ) h., m., m,..., h, h+1 : { [dk, η] k h, d k k 1, η Υ m k+1 (Q m k+1 ) } { [ µ h 1 L (l), Υ 2(x j )] l q, j q }. ImΥ h, (N h,l )

7.1. (8) p 1 h (N h,l) Υ h+1 (Q h+1 ) h., p 1 h (N h,l) Υ h+1 (Q h+1 )., µ-., : 7.2. 5 2 1, 7 2 6, 7 2 8, 8 2 13, m = 4 ( 1 ).. h = 5, µ 5 L (j) ±[[[[λ(2) 1, λ (1) 4 ], λ (2) 5 ]. (, µ-)., h = 6, 7, h.,,,., #L 3, #L = 2 lk(l) 2.,.. 7.3. L Borromean 6 3 2. 5.1 µ3 L (j) [[λ(j) 1, λ (j+1) 2 ], λ (j+2) 3 ] ( j Z /3), 4 : 4 = Z [[[λ (j) 1, λ (j+1) 2 ], λ (j+2) 3 ], λ (k) 4 ] j,k Z /3., h = 4,. µ 4 L(j) [[[λ (j) 1, λ (j+1) 2 ], λ (j+1) 3 ], λ (j+2) 4 ], modulo 4., h = 5, 5 : [[[[λ (j) 1, λ (j+1) 2 ], λ (j+2) 3 ], λ (k) 4 ], λ (l) 5 ], [[[λ (j) 1, λ (j+1) 2 ], λ (j+2) 3 ], [λ (k) 4, λ (l) 5 ]], [ µ 4 L(j), λ (k) 5 ] j,k,l Z /3., h = 5 : µ 5 L(j) [[[[λ (j) 1, λ (j+1) 2 ], λ (j+1) 3 ], λ (j+1) 4 ], λ (j+2) 5 ], modulo 5., h with k = 3, 4, k+1 modulo µ k+1 L (j).,, µ h L (j) h,., c(l) < 11 Borromean. L = 9 3 n25 L = 10 3 a151. : µ 4 L (1) [[[λ(2) 1, λ (3) 2 ], λ (3) 4 ]] [[[λ (3) 4 ]] [[[λ (3) 1, λ (1) 3 ], λ (3) 4 ], µ 4 L (2) [[[λ(2) 1, λ (3) 2 ], λ (3) 4 ], µ 4 L (3) [[[λ(3) 1, λ (1) 3 ], λ (3) 4 ], µ 4 L (1) [[[λ(2) 1, λ (3) 2 ], λ (3) 4 ]] [[[λ (1) 4 ], µ 4 L (2) [[[λ(3) 1, λ (1) 4 + λ (2) 4 ] [[[λ (3) 3 ], λ (3) 4 ], µ 4 L (3) [[[λ(1) 3 ], λ (3) 4 ]., #L 3 µ-., L = 6 3 2 L = 9 3 n25 Alexander. [Mu2] µ-,..

7.4., #L = 2., lk(l) = 0., 7.5. #L = 2. L µ-, lk(l) Z.., lk(l)[λ (1) 2 ], 7.1, h with h > 2 lk(l) Z ; µ-.,, lk(l) 3., #L = 2 lk(l) = 3., c(l) 9 lk(l) = 3.., 4, µ-. Link L µ 3 L (j) µ4 L (j) 5 µ 5 L (1) µ5 L (2) 6 2 1 0 2b 1 + b 2 + b 3 3, B + D F, A + C + E A C A + C + D 6 2 2 0 2b 1 + b 2 3, B E, A C D F B F 8 2 a10 0 2b 1 + b 2 b 3 3, B D + F, A C E C C 8 2 a11 0 2b 1 b 2 + b 3 3, B + D F, A + C + E C C 9 2 a23 0 2b 1 + b 2 + b 3 3, B + D F, A + C + E A + B + D A + C D 9 2 a28 0 2b 1 + b 2 3, B E, A C D F B D F 9 2 a32 0 2b 1 + b 2 + b 3 3, B + D F, A + C + E A + C A C 9 2 a33 0 2b 1 + b 2 b 3 3, B D + F, A C E A + B + C A + B + C 9 2 n15 0 2b 1 + b 2 + b 3 3, B + D F, A + C + E A C + D A + C D 9 2 n16 0 2b 1 + b 2 + b 3 3, B + D F, A + C + E A C + D B C + D, j {1, 2}, : b 1 = [[[λ (1) 3 ], λ (2) 4 ], b 2 = [[[λ (2) 1, λ (1) 4 ], b 3 = [[[λ (1) 4 ], A = [[[[λ (1) 3 ], λ (2) 4 ], λ (2) 5 ], B = [[[[λ (1) 3 ], λ (2) 4 ], λ (1) 5 ], C = [[[[λ (2) 1, λ (1) 4 ], λ (2) 5 ], D = [[[[λ (2) 1, λ (1) 4 ], λ (1) 5 ], E = [[[[λ (2) 1, λ (1) 3 ], λ (2) 4 ], λ (2) 5 ], F = [[[[λ (1) 4 ], λ (1) 5 ]. A : µ-, ([M1, IO, Hil] )., Υ m,. Υ m Taylor ( A.1)., Fox. k {1,..., q}, x k : F Z[F ] () : x i x k = δ i,k, (uv) x k = u x k ε(v) + u v x k, for all u, v F., ε Z[F ] Z.,, y F, n y = ( n 1 y ) x i1 x in x i1 x i2 x in. D i1 i n (y).

A.1. y F, Υ m (y) : 1 ε (D k1 (y)) λ (k1) 1 ε (D k1k 2 (y)) λ (k1) 1 λ (k2) 2 ε ( D k1 k m 1 (y) ) λ (k1) 1 λ (km 1) m 1 k 1 k 1,k 2 k 1,...,k m 1 0 1 ε (D k2 (y)) λ (k2) 2 ε ( D k2 k m 1 (y) ) λ (k2) 2 λ (km 1) m 1 k 2 k 2,...,k m 1 0 0 1 ε ( D k3 k m 1 (y) ) λ (k3) 3 λ (km 1) m 1 k 3,...,k m 1......... 0 0 0 0 1 ε ( D km 1 (y) ) λ (km 1) m 1 k m 1 0 0 0 0 1, k s, k s+1..., k t, {1, 2,..., q} t s+1., y,.,, Υ m : F/Γ m F GL m (Ω m ) (cf. Taylor )., (9), [CFL, 2]. (c 1 c 2 c k ) {1,..., n} k I = a 1 a 2 a I J = b 1 b 2 b J. (c 1 c 2 c k ) I J, I α(1), α(2),, α( I ) J β(1), β(2),..., β( J ), : (i) 1 α(1) < α(2) < < α( I ) k, 1 β(1) < β(2) < < β( J ) k. (ii) i {1, 2,..., I } j {1, 2,..., J } c α(i) = a i c β(j) = b j. (iii) s {1, 2,..., k}, α(i), β(j),. I J, Sh(I, J) (, Sh(I, J) )., [CFL, Lemma 3.3], multi-indexes I J, y F, : ε(d I (y)) ε(d J (y)) = ε(d K (y)) Z. (9) K Sh(I,J), ( m )Magnus. Z X 1,..., X q, X 1,..., X q, J m m., F Magnus, M : F Z X 1,..., X q /J m, : M(y) = ε(y) + ε(d i1 i n (y)) X i1 X i2 X in. (10) n: 0 n<m (i 1,...,i n) {1,2,...,q} n, M, M(Γ m F ) = 0., Γ m F : M : F/Γ m F Z X 1,..., X m /J m., [CFL, Theorem 3.9], : { a I X i1 X in J and K, a J a K = } a L Z. (11) I=(i 1 i n) L Sh(J,K), A.1, 1 + X i Υ m (x i ) ImM Im(Υ m )., (11), Im(Υ m ) GL m (Ω m ).

B., : [CEGS, 5]., (12). A.1, Im(Υ m )., s : Im(Υ m ) Im(Υ m+1 ) (, s, (1, m + 1)-)., ϵ {±1},. ϕ ϵ m : Im(Υ m ) Im(Υ m ) GL m+1 (Ω m+1 ) (12) s. : { ϕm(a, ϵ s(b) 1 s(a)s(b)s(b 1 AB) 1, if ϵ = 1, B) = s(b)s(a)s(b) 1 s(bab 1 ) 1, if ϵ = 1., p m ϕ ϵ m, ϕϵ m Ker(p m) = Υ m+1 (Q m+1 )., B.1. D A m. 4, α k, β k, ϵ k, 1,, Wirtinger, f m { arc of D } Im(Υ m )., (12), Φ m,j (k) := ϕm( ϵt fm (α t ), f m (β t ) ) Υ m+1 (Q m+1 ). (13) t: 1 t k. [CEGS], Φ m,j (k). B.1 ([CEGS, 5] )., Φ m,j (k), (3) Ψ m (j).,, Q m, ( ), Φ m,j (k)., Ψ m,j (k), Φ m,j (k). Φ m,j (k)., µ- ( ). [CEGS] J. S. Carter, J. S. Elhamdadi, M. Graña, M. Saito, Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math. 42 (2005), 499 541. [CFL] K. T. Chen, R. H. Fox, R. C. Lyndon, Free differential calculus IV, the quotient groups of the lower central series, Ann. of Math. 68 (1958), 81 95. [GG] C. K. Gupta, N. D. Gupta, Generalized Magnus embeddings and some applications, Math. Z. 160 (1978), 75 87. [HM] K. Habiro, J.-B. Meilhan, Finite type invariants and Milnor invariants for Brunnian links, Int. J. Math. 19 (2008), 747 766. [Hil] J. Hillman, Algebraic invariants of links, Series on Knots and everything. 32 World Scientiffc (2012). [IO] K. Igusa, K. Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001), 1125 1166. [KN] H. Kodani, T. Nosaka, Milnor invariants via unipotent Magnus embeddings, preprint. [M1] J. W. Milnor, Link groups, Ann. of Math. 59 (1954) 177 195. [M2], Isotopy of links, in Algebraic geometry and topology. A symposium in honor of S. Lefschetz, 280 306, Princeton University Press, Princeton, NJ, 1957 [Mu1] K. Murasugi, Nilpotent coverings of links and Milnor s invariant, Low-dimensional topology, London Math. Soc. Lecture Note Ser., 95, Cambridge Univ. Press, Cambridge-New York (1985), 106 142. [Mu2] K. Murasugi, On Milnor s invariant for links, Trans. Amer. Math. Soc. 124 (1966), 94 110. [St] D. Stein, Computing Massey product invariants of links, Topology and its Applications, 32 (1989), 169 181.