2.1.,., { n Q[t ±1 ] := a k t k a k Q, m, n N k= m. Z., s Z, n k= m a kt k s := n k= m a kt k+s. : Q[t ±1 ] {t n } n Z Q t 2 Q t 1 Q t 0 Q t Q t 2 (Q-

Transcription

1 ( ) 2018,.. 1, 1 H 1 (X; Q) 0 1 CW X.,,...,,..,,. [Adams, CF, Lic,,, Rol] [Gor,, Kaw],,,, (, ).,,.,,.,,. (3 )., ( ) (, ).. 3,.,,.,,.,., Blanchfield.,. 2.,,,. 1 H 1 (X; Q) 0,. 3,, H 1 (X; Q) = 0. 1

2 2.1.,., { n Q[t ±1 ] := a k t k a k Q, m, n N k= m. Z., s Z, n k= m a kt k s := n k= m a kt k+s. : Q[t ±1 ] {t n } n Z Q t 2 Q t 1 Q t 0 Q t Q t 2 (Q- ),,. (, Q[t ±1 ]-, Q Z ). }., n- f = n k=0 a kt k Q[t], Q[t ±1 ]/(f) ( a n 0, a 0 0 ). n, 1, t, t 2,..., t n 1. 2, Q[t ±1 ]/(f).,. n 2,., t n 1 1 t, t t 2,..., t n 2 t n 1, t n 1 t n (a 0 + a 1 t + + a n 1 t n 1 )/a n. (f) 3., n V Z (, Q[t ±1 ]- ). V v 0, n, v 0 s t s., V 1, t,..., t n 1., t n 1, a 0,..., a n 1, t n 1 1 = a 0 a 1 t a n 1 t n 1., f = a 0 + a 1 t + + a n 1 t n 1 + t n, V Q[t ±1 ]/(f).,.,. Q[t ±1 ] W 1 = 1 Q t i i Z V,, f 1, W 1 = Q[t]/(f1 )., V/W 1, W 2 = Q[t]/(f 2 )., : V = W 1 W m = Q[t]/(f1 ) Q[t]/(f m ), 2.1 ( ). V, Z. V Q[t ±1 ]. f 1,..., f m 4, Q[t ±1 ]. V = Q[t]/(f 1 ) Q[t]/(f 2 ) Q[t]/(f m ). (1) 2 Q[t ±1 ] ( ) ( ). 3,, ( ). 4, f k, a Q t ±1 ( (Q[t ±1 ]) = {at n } a Q,n Z.) 2

3 , Z V, f 1,..., f m.,. 2.2 M ( ). α : π 1 (M) Z, M M. π 1 (M). M, π 1 (M) := { f : S 1 M f, f(0) = }/.,., ( ) ( [, ] ), p : M M, ( ) p (π 1 ( M)) π 1 (M) : { M M}/ {N π 1 (M) N.}.,.,,.,. α : π 1 (M) Z, Ker(α), Y CW, [Y, S 1 ] K S 1 ( S 1 = R/Z ). [ϕ] [Y, S 1 ], γ(ϕ) : π 1 (Y ) π 1 (S 1 ) = Z. γ : γ : [Y, S 1 ] Hom(π 1 (Y ), Z) 5. Proof... Y, Y k k-., f : π 1 (Y ) Z, ˆf : Y 1 S 1. π k (S 1 ) = 0 (k > 1), ˆf Y 2-skeleton., Y S 1., up to ( ),.. [, 6.2.5]., α α : M S 1.,. exp:r S 1 y exp(2π 1y)., X R ( ) M := {(x, y) X R α(x) = exp(y) }.. p(x, y) = x, α(x, y) = y,. 5, Hom(π 1 (M), Z) = H 1 (M; Z).. 3

4 M M α R ˆα S 1 1: S 1. M t : M M t(x, y) = (x, y + 1). M ( ) p : M M {t n } n Z,., x X, O Y, M Õ (1)(2) : (1) n, pt n = p p t n Õ : tn Õ O. (2) p 1 (O) = n Z t n Õ. Õ, t±1 Õ, t ±2 Õ. Proof. pt n = p. x M, α(x) V S 1, p 1 (V ) = n Z t n Ṽ Ṽ R. exp Ṽ : Ṽ V, O = α 1 (V ), Õ = α 1 (Ṽ ),, (1)(2) α,, M. ([, p.177] )., M, α. 2.3 M Z = {t n } n Z ( )., H ( M; Q). Q[t ±1 ] ( ). V H k ( M; Q), Q[t ±1 ], (1). f 1 f m (α k ). k (t) (k = 1, k ) H k ( M; Z)( Z[t ± ], ). 2.7.,, Q t ±1, (, (f i ) f i, ).,, 6. f 1 f m 0, H k ( M; Q) Q.. : 2.8 ( [, 6.2.8]). H ( M; Q), M χ(m).,.,. 4

5 2.9. : M = S 1 S n, M = R S n,, H n ( M; Q) = Q[t]/(t 1),. M. H 1 (M; Z) = Z Z/2. M S 1, H 0 ( M; Q) = Q[t]/(t 1) H 1 ( M; Q) = Q[t]/(t + 1), n 2, M S 1 S n. M S n ( 2 ), H ( M; Q) Q[t ±1 ],., M S 1 S 2n 1 S 2n, H ( M; Q) (χ(m) = 0 ). 2: S n, S 1 S n,, : f : M S 1 fibered. dim(m) 1 W, f : W W, M = (W [0, 1])/{(x, 0) (f(x), 1)} x W 6. f : π 1 (W ) π 1 (S 1 ) = Z α., M W R. W, H ( M; Q). dim(m) = 3, W Σ g,r, H 1 (W ; Z) Z 2g+r f : H k (W ; Q) H k (W ; Q) A, Alexander Det(tA I rkhk (W ;Q)) (Hint: ) H (M; Q) H (S 1 ; Q) 7., H ( M; Q). Proof. M 1 = t, M. 0 C cell ( M; Q) 1 t. C cell ( M; Q) C cell (M; Q) 0 (exact) Hn+1(M; cell Q) Hn cell ( M; Q) 1 t Hn cell ( M; Q) Hn cell (M; Q) (2) ( Wang ). n > 1,, 1 t. Hn cell ( M), Q[t ±1 ], Q[t ±1 ] 1 t., n > 1 Hn cell ( M)., n = 1, 0, 0 H 1 ( M; Q) 1 t H 1 ( M; Q) H 1 (M; Q) H 0 ( M; Q) 1 t H 0 ( M; Q) H 0 (M; Q) 0 6. f, x S 1, f 1 (x) W. 7, K : S n 2 S n, M = S n \ Im(K), Alexander H (M; Z) H (S 1 ; Z) 5

6 ., 4 Q., 1 t., n 1 Hn cell ( M; Q) H ( M; Q)., (2) χ( M) = χ( M) + χ(m) χ(m) = 0., Alexander 3,., M M R. m N, M (x, y) (x, y + m) m, M m., M s S 1 S n 2, M m ( M = S 3 \ K )., M m s S 1 D n 1, m ( ). B m., M M m B m H 1 ( M; Z) H 1 (B m ; Z), (t m 1)/(t 1)H 1 ( M; Z)., : H 1 (B m ; Z) = H 1 ( M; Z) (t m 1)/(t 1)H 1 ( M; Z) Z[t ±1 ]. [Kaw, 5.5]. [Lic, Cor 9.8] : 3.2. M = S 3 \ K, Alexander (t 1 ) = (t), (1) = ±1., m, : m 1 H 1 (B m ; Z) = (e 2π 1v/m )., 3.3 ( - [NY]). M S 3 \ K. v=1 {f Hom(π 1 (M), SU(2)) f } SU(2), ( 1 ( 1) + 1)/2. 6

7 4 Alexander,.. Z,., Alexander., Alexander. 4.1 Z[t ± ] order.,, Q[t ±1 ]., Z. H ( M; Z) Z[t ±1 ]., Z[t ±1 ] ( )., Z[t ±1 ] M ( M < ).,, Alexander order. S, A S., s, r N S r P S s A 0 ( ), (P s r )., i E i (A), : { det(b) S B : (s i) (s i) P }. ( ) a 11 a 12 a P, : a 21 a 22 a 23 a 11 a 12 E 0 = 0, E 1 = a 21 a 22, a 11 a 13 a 21 a 23, a 12 a 13, E a 22 a 23 2 = a ij i 2,j 3, E 3 = A., A E i (A) ( [CF] ) ( [CF] ), S UFD( ) 8.,, E 0 (A) S. S S order, ord(a) S. S S, ( ). Order,, : 4.4. ( Q(S) S, Tor S A A ): orda 0 E 0 (A) 0 Q(S) S A = 0 ranka = 0 A = Tor S A S PID ( Z, Q[t ±1 ]). M S,, m, n N f 1,..., f m S, S M = S m ( S/(fi ) ) 1 i n 8 R x x = p 1 p 2 p n R R., Z[t ±1 1,..., t±1 m ]. 7

8 . S n+m S n+m M 0, P = diag[0,... (m )..., 0, f 1,..., f n ].., E 0 = det(p )S., ord(a)=±det(p )S. : 4.6., M. H k ( M; Z) Z[t ±1 ]-. order Z[t ±1 ], (k )Alexander ( ±t m± ). 2.6 Q[t ±1 ], : 4.8. M M, H ( M; Q). Z[t ±1 ] (k )Alexander. Q[t ±1 ], 2.6 (k )Alexander. Proof. S = Z[t ±1 ], S n S m H ( M; Z) 0. Q, H ( M; Q). Q order. 4.5 order f 1 f m, 2.6 (k )Alexander H ( M; Q), Alexander. 4.2 Alexander., Alexander.., 7, Alexander. Alexander., Alexander., M 2k + 1., k Alexander (reciprocity)., k (t 1 ) = t deg k (t). k = a m t m + + a 0 t a m t m = a i = a i., H s (M; Z) H s+1 (M; Z)., Alexander t = 1, s (1) = ±1. 8

9 9, 6.,., : 4.11 ([Yj]). n > 1. f Z[t ±1 ], K : S n S n+2, H 1 ( Sn+2 \ K; Z) = Z[t]/(f). Alexander fibered : M 3, M 2.11, fibered., Alexander ±1. H 1 (Σ g,r ; Z)., 10., K : S 1 S Alexander ±1, K fibered. Slice. ( )Slice, F : D 2 D 4, (ImF ) D 4 = S 3, F D 2 = K. Alexander ( ) K : S 1 S 3 Slice, g(t) Z[t ±1 ], (t) = g(t 1 )g(t). [, 5.3.4] [Lic, 8.18]., K S 3, (K ). g (K) := min{g Z K g. } : deg 1 (t) 2g (K). g (K),, deg 1 (t) = degdet(tv V T ) rkv = 2g (K)., Knot Floer homology, g (K) [OS]. 9 Reidemeister [, 5.3]. [CF, IX ]. Blanchfield pairing [Gor, Kaw, Hil].,. 10 Alexander, Knot Floer homology, homology fibered-ness [Ni]. 11,, arc,. 9

10 5 Alexander H 1 (X) = Z, 1 ( )., π 1 (M), π 1 ( M)., H 1 ( M; Z) = π 1 (M) /[π 1 (M), π 1 (M) ]. Z = t. Z[t ±1 ], Z[t ±1 ].., M 2.11 π 1 (M) = π1 (W ), t,, : 5.1., M := S 3 \ K 31. G := π 1 (S 3 \ K 31 ). G := π 1 (S 3 \ K 31 ) = x, y yxyx 1 y 1 x 1. π 1 (S 3 \ K 31 ) Z x 1, y 1. a = yx 1 G = x, a ax 2 ax 1 a 1 x 1 = x, a a(x 2 zx 2 )(xa 1 x 1 ), π 1 ( M) = G : H 1 ( M; Z) = G /G = α, t α + t 2 α tα. t 2 t [Rol, 7.D],, M = S 3 \ T (p, q) (p, q ): S 3 \ T (p, q) := {(x, y) C x 2 + y 2 = 1, x p + y q = 0 }., 1 (t) = (1 t)(1 tpq ) (1 t p )(1 t q ), H 1( M; Z) = Z[t ±1 ]/ 1 (t).,., π 1 (M) /[π 1 (M), π 1 (M) ], t 2 t + 1 t 2 3t + 1 t 5 1 t 1 2t 2 3t + 2 3:. 12, 5 2, H 1 ( M; Z) = Z[t ±1 ]/(2t 2 3t + 2) Z. 10

11 5.2 Alexander 2. Alexander., ( ). S 1,., L : m S 1 R 3. m #L. L L, h : R 3 [0, 1] R 3, h t R 3, h 0 = id R 3 h 1 (L) = L., D : m S 1 R 2,,. 7 L +, 7 L,. L, p : R 3 R 2, p L., 4., : { } 1:1 { }. RI RI RII RII RIII 4:.,. D π 1 (S 3 \L). Wirtinger. D arc,. 11

12 5.4 (Wirtinger ( [Rol])). π 1 (S 3 \ L). e α (α D arc ) τ, e 1 γ τ e 1 β τ e ατ e βτ ), e α, α.. α τ β τ β τ γ τ γ τ α τ,.,.,, H 1 (S 3 \ L; Z) Z #L. Proof. i #L, α, γ L, Wirtinger,. α, π 1 (S 3 \ L) Z #L..,,. L, D, D = D 1 D #L. p, 2. p, ϵ(p) = +1, p, ϵ(p) = 1. (i j), i j. lk(d i, D j ) := 1 ε(p). 2 p D i D j.. [Rol, 5.D ] #L #L {link(l i, L j )} i,j #L ( Jordan ).,,. 5.3 FOX Alexander. Alexander.. G Z[G]., Z[G] := { a i g i a i Z, g i G }.., H := G Z k ( m i=1(z/n i Z)), Z[H] : Z[H] = Z[t ±1 1,..., t ±1 k, s 1,..., s m ]/(s n 1 1 1,..., s n m m 1). 12

13 Z[H] UFD, H., FOX F I I. k I x k. FOX( ), k I, x k : F I Z[F I ]. x i x k = δ i,k, (uv) x k = u v x k + u x k, for all u, v F I F I = x, y, yxy x yx y + y x + 1 = yx + 1. y y 5.9. well-definedness ((uv)w) w 1 x i 1 w = w x i wn x i = y xy x + y x x k = (u(vw)) x k = yx y x + y x x. = y, yxy y = (1 + w + + w n 1 ) w x i (n > 0). w F, w 1 = s i=1 w x i (x i 1). = y xy y + y y =, Alexander. G = x 1,..., x n r 1,..., r m, H G, a : G H. Alexander, G r 1,..., r m, r i x j,., Z[H] n m a( r 1 x 1 ) a( r 1 x n ) A G :=..... Mat(n, m, Z[H]). a( r m x 1 ) a( r m x n ) A G k, k k, x 1,..., x n r 1,..., r m ( Tietz ). ( [CF, 4.5] ) H, Z[H] UFD, order., Alexander :, G := π 1 (S 3 \ L). H = Z #L 5.5. A G k order, (k ) Alexander. 1 Alexander G ( G H = Z, Z[H] = Z[t ±1 ] ), k order, 5.1 k Alexander ( ) G = x, y xyx = yxy 3 1. : α( x (xyxy 1 x 1 y 1 )) = α(1 + xy xyxy 1 x 1 ) = 1 + t 2 t. 13

14 α( x (xyxy 1 x 1 y 1 )) = α(x + xyxy 1 xyxy 1 x 1 y 1 ) = 1 t 2 + t. Alexander (1 t + t 2 1 t 2 + t). order, Alexander 1 t + t , Alexander., 8 18, t 2 t T (p, q), x, y x p = y q,, FOX, Alexander. FOX, Alexander. 9 CW.., CW X, p : X X., X π1 (X). X CW., X j- σ j, 1 σ j, p 1 (σ j ) π 1 (X) σ j., C j ( X; Z) = Z[π 1 (X)] σ j (3) σ j :X j. X C : C k ( X; Z) k Ck 1 ( X; Z) k 1 C1 ( X; Z) 1 C 0 ( X; Z) 0, k π 1 (X)., M, π 1 (X) M,. H n (C ( X; Z) Z[π1 (X)] M, k id) ab : π 1 (X) H, Ker(ab) X ( ). M Z[H],, C ( X; Z) Z[π1 (X)] M, X C (X; Z)., k,. 2., X CW, 0-cell 1, 1-cell n, 2-cell m., π 1 (X) x 1,..., x n r 1,..., r m ( 9 )., X i-cellσ, p 1 (σ) π 1 (X) i-cell., C 1 ( X; Z) Z[π 1 (X)] n, C 2 ( X; Z) Z[π 1 (X)] m. FOX, : C 1 ( X; Z) 1 C 0 ( X; Z) = Z[π 1 (X)] x k 1 x k., C 2 ( X; Z) 1 C 1 ( X; Z) FOX., 5.17, M = Z[H], Alexander, C 2 (X; Z) 1 C 1 (X; Z). 14

15 ,. H 1 (X; Z) X,, Alexander. 6, Alexander. [Rol, 8 ] [Lic, 6 ] (,, ). L S 3.. α : π 1 (S 3 \ L) Abel Z #L Z.,. 6.1, : 6.1. L S 3. L, S 3 \ L Σ, ( )L., Σ S L, , ([Lic, 8 ] ). Proof. L D. ˆD 5 D., ˆD,. ˆD R 2. ˆD,.,.,, ˆD, L.,,,. ( ). 2.3 α C - S 3 \ L S , α 1 (0) (S 3 \ L) #L. 0, Σ := α 1 (0),. Σ,. = = 5: [Lic, 5.A]. google.,. 15

16 K 41, K 31 K 41, (,.) 6.5. Σ S 3, N(Σ)., S 3 \ N(Σ) 13. ( 6, 1 g attach. 2. m S 1. ) ( 6.12), Alexander., (Alexander ). 6.6 ([Lic, 6.3] ). Σ S 3 g Seifert., H 1 (S 3 \ F ; Z) H 1 (F ; Z) (Z 2g+#L 1 )., β : H 1 (S 3 \ F ; Z) H 1 (F ; Z) Z, F S 3 \ F c d, β([c], [d]) = lk(c, d).. V S 3, V S 3 \F. V V = S 3 V V V, Mayer -Vieoris, H 2 (S 3 ; Z) H 1 ( V ; Z) H 1 (V ; Z) H 1 ( V ; Z) H 1 (S 3 ; Z).,. H 1. 6,., H 1 (V ; Z) = H 1 (F ; Z) {f i } i 2g+#L 1,., H 1 (V ; Z) e i, 6 (f i ). H 1 (V ; Z)., V,, H 1 ( V ; Z) = Z 2g+#L 1 Z 2g+#L 1,., β([e i ], [f j ]) = δ ij β., 6 β([c], [d]) = lk(c, d) ( ).. e 1 e 2 e 4 e2g 1 e 2g f 1 f 2 f 3 f 4 f 2g 1 f 2g f 2g+1 f 2g+2 f 2g+#L 1 6: H 1 [f j ], H 1 (S 3 \ F ) [OSJS, 3 4 ],, trefoil (3 ), 3, m S 1. Σ m R

17 ,., [0, 1],. + : F {0} F {1} 6.8. L S 3, F. F, : H 1 (F ; Z) H 1 (F ; Z) Z; ([α], [β]) lk(α, (β) + ). α, β H 1 (F ; Z) F well-defined [Lic, 6.3]..,,., : Z[t ±1 ] Λ ( ). tv T V H 1 ( S 3 \ L; Z)., Λ 2g tv T V Λ 2g H 1 ( S 3 \ L; Z) 0 (exact)., F,. F, F ( 1, 1) S 3 \ F. Y S 3 \ F ( 1, 1), F = F { 1} F + = F {1} K ( 1, 1). { 1} {1} ϕ : F F +., i Z, Y Y i. h i : Y i Y i+1. i Z Y i X = i Z Y i / {(x, i) (ϕ h i (x), i + 1)} i Z, x F+.. t : X X t Yi := h i+1 h 1 i. Y i X S 3 \ K α. 17

18 2.4., α, [Lic, 7.9] ( ) ( [Lic, 6.5] ). X :Y := i Z Y 2i+1, Y := i Z Y 2i., Y Y X, Mayer-Vietoris H 1 (Y Y ) H 1 (Y ) H 1 (Y ) H 1 (X ) δ H 0 (Y Y ) H 0 (Y ) H 0 (Y ) Z, Y Y F,, δ. H 1. H 1 (Y Y ) Z[t ±1 ] H 1 (F ; Z). 6.6 H 1 (S 3 \ F ; Z) = H 1 (F ; Z), H 1 (Y ) (Z[t ±2 ]) 2g+#L 1, H 1 (Y ) (tz[t ±2 ]) 2g+#L 1.,. (Z[t ±1 ]) 2g+#L 1 α (Z[t ±1 ]) 2g+#L 1 H 1 (X ) 0 (exact).. V ij ij., Mayer- Viertoris, 6.6, α(1 [f i ]) = ( V ij (1 [e j ]) + V ji (t [e j ])) j rkv ( ).,,., : Alexander. K (t) := det(t 1/2 V T t 1/2 V ) Z[t ±1/2 ]. 4.7 Alexander,, : 6.13., 4.7,,., Alexander K (t 1 ) = K (t).,, t = 1, K (1) = 1. Proof. K (t 1 ) = det(t 1/2 V T t 1/2 V ) = ( 1) rkv det(t 1/2 V T t 1/2 V ) = K (t)... {f i },. K (1) = det(v T V ), ij (V V T ) ij = lk(f i, f j) lk(f + i, f j) ( 0 1, f i f j. 1 0., det 1. ) g 18

19 6.15. Alexander order, ±t ±1/2., K, K (t 1 ) = K (t) (1) = 1,., 6.12 Seifert, (±t ±1/2 ) ([Lic, 8] )., S k S k+2., k ([Hil, Rol] ) 6.2 Seifert. 5.13, Fox 5.12, 4.7., S 3 \ K,., π 1 (S 3 \ K) (4). ([Lin, 2] ). 1- H S 3 K m π 1 (S 3 \ K), g Seifert F S 3 K. U := S 3 F, U := H (F )., U U F F. 6.5, U (resp. U U U ) 2g (resp. 2g + 1 4g). 6, e i f i (1 i 2g)., π 1 (U ), {(f 1 ) +,..., (f 2g ) + }., π 1 (U ) {(f 1 ),..., (f 2g ), m} π 1 (U U ) {(f 1 ) +,..., (f 2g ) +, (f 1 ),..., (f 2g ) }., {e 1,..., e 2g } π 1 (U ). 6.6, e i f j δ ij. i : U U U i : U U U., van-kampen π 1 (S 3 \ K) m, e 1, e 2,..., e 2g, ma j m 1 b 1 j, (1 j 2g), (4) a j b j {e 1,..., e 2g }., π L = π 1 (U U ) {f 1,..., f 2g, (e 1 ),..., (e 2g ), } i (e j )i (e j ) 1 = m i ((f j ) + ) m 1 (i ((f j ) )) 1, (e 1 ),..., (e 2g ) i (e j )i ((e j ) + ) 1 ) ( )., [Lin, ]., : π 1 (S 3 \ 3 1 ) = x 1, x 2, h hx 1 x 1 2 h 1 = x 1, hx 2 h 1 = x 2 x 1 1. Trotter X ( ) 6.17 ( [Tro, 4.1])., X. 0 Z[t ±1 ] 2g (V T tv, w) Z[t ±1 ] 2g+1 1 t Z[t ±1 ϵ ] Z 0 (exact)., w. 19

20 ., 1 e (i) j 1 t, X., a (i) j X. i Z, e j e (i) j, e j e (i) j. b (i) j Shreirer, π 1 (X ) : e (i) 1, e (i) 2,..., e (i) 2g, (i Z) a (i+1) j.,., Reidemeister- (b (i) j ) 1, (1 j 2g) (i Z). r (i) j, X 2-cell., 2 (r (i) j ), e j e j, e j e + j. 6.10, α., α., ( ma jm 1 b 1 j w. m), 2 α = (V T tv, w)., , Alexander π K, π K. 6.14, Z[t ±1 ] (1 t) 1 Z[t ±1, (1 t) 1 ]. 5.18, 2 (4) Alexander., 6.17, 1 w, 2 k- order, V T tv (k 1)- order., 6.10, V T tv H 1 (X ; Z)[(1 t) 1 ]., Alexander. Alexander A : { } Z[t ± 1 2 ]. D 1, D 2 Reidemeister, A(D 1 ) = A(D 2 ). U, A(U) = 1. 3 L +, L, L 0, 1, 7,,, ( ) : A(L + ) A(L ) = (t 1/2 t 1/2 )A(L 0 )., 6.19., S 3 \ K, A(K) 1 (t). [Lic, 15.2], [Lic, 8.6] [, 3.2].,.,.,,. 14.,, X, π 1 (X ), X., 2 π 1 (X ),. X 3, 3. 20

21 Alexander,. L 0 F 0 8 ( 6.2 )., L + L 8. V 0, V +, V., H 1 (F 0 ; Z), {f 2,..., f n }. f 1 8, {f 1, f 2,..., f n } H 1 (F ±1 ; Z).,, V + V : M + = ( M0 w v T n ), M = ( M0 w v T n + 1 v w.,. det(tv T + V + ) det(tv T V ) = (1 t)det(tv T 0 V 0 ). t rkv +/2, L (t),. ). 6.4., Colored Jones (Melvin-Morton ) [OSJS, 5 ]. L + L 0 L 7: 21

22 f 1 f 1 8:, f 1. 22

23 7 Reidemeister 7.1.,. (i) Reidemeister. 3. (ii) Franz,. (iii) Whitehead,. Whitehead. (iv) Milnor [Mil4], Alexander. (v) Turaev[Tu1, Tu2], refine., Spin c, ([Nic] )., ,. ([Tu3] ) (vii) 2000,,. ( [FV] ),., [Tu1, Mas], [, ]. [Nic, Tu2] ( Turaev ) 7.2,. F, F : C : 0 C m m 1 m Cm C1 C0 0, ( ). (5), C q c q := {c (q) 1,..., c (q) n q }., ( ).,. ( ). C q a q := {a 1,..., a m } b q := {b 1,..., b m }. a i = m j=1 d ijb j {d ij }. det{d ij } [a q /b q ] m = 1., 0 C 1 C 0 0, 1 : C 1 C 0. ( ). det( 1 ) m = 2, 0 M (f,0) M N (0,g)T N 0. f, g., detf/detg. 15, Euler class, Casson, LMO, Seiberg-Witten 23

24 q+1 q , 0 C q+1 Cq Cq 1 0. Im( q+1 ) = B q C q., B q b q. : q+1 0 C q+1 Bq 0, 0 Z q C q B q 1 0, 0 B q 1 C q 1 0. F, s : B q 1 C q,,. q+1 0 Z q Im( q+1 ) Im(s) B q 1 0. bq 1 s(c q 1 ).., c q+1 b q 1, C q., [c q+1 bq 1 /c q ]. s, ( ). 3 : [b q+1 /c q+1 ] [ b q 1 /c q 1 ]., : [b q+1 /c q+1 ] 1 [b q bq 1 /c q ][ b q 1 /c q 1 ] (Reidemeister ). (5),. S q : 0 Z q (C ) C q B q 1 (C ) 0 Reidemeister, τ(c ),. m τ(c, c q ) := [b q, b q 1 /c q ] ( 1)q+1 F \ 0 q=0 7.5., B q. Proof. {b q}, [b q b q 1/c q ] = [b q bq 1 /c q ][b q/b q ][b q 1/b q 1 ]., m m [b q, b q 1/c q ] ( 1)q+1 = ([b q bq 1 /c q ][b q/b q ][b q 1/b q 1 ]) ( 1)q+1 q=0 = q=0 m ([b q bq 1 /c q ]) ( 1)q+1 ([b q/b q ][b q 1/b q 1 ]) ( 1)q+1 = q=0 m [b q, b q 1 /c q ] ( 1)q+1. q=0 7.6., τ(c ) C., c q, ( [, Lemma 5.1.6]): m τ(c, c q ) = τ(c, c q) [c q /c q] ( 1)q+1., c q, c q. q=0 24

25 ,., C 3 C 2 0 C 1 C 0 0 ( 0), det( 3 )det( 1 ) A m A m+1 A 0, 2 (A, w), 1 = ( v, b) T, w., A m m.,, det(a)/b ( )., : C C C 0., c q, c q, c q C q, C q, C q, ι : C i C i p : C C i,. [ι(c q)c q/c q ] = 1., τ(c) = ±τ(c )τ(c ) , ([ ] ). 7.3 CW Reidemeister. 5.3, π 1 (X) = G H = π 1 (X) ab, X. φ : π 1 (X) F., C : C k ( X; Z) Z[π1 (X)]F k C1 ( X; Z) Z[π1 (X)]F 1 C 0 ( X; Z) Z[π1 (X)]F (Reidemeister )., τ φ (X) = 0.,,, φ(π 1 (X)) : τ φ (X) := τ(c ( X Z[π1 (X)] F)) F /φ(π 1 (X)),., CW,., Reidemeister. [Tu1, 8.8] ( 16 ).,,, 16 : simple homotopy.. simple homotopy. 25

26 7.14.., Reidemeister. (, ) (Chapman[]). CW h : X X., Reidemeister τ φ (X) = τ φ h (X )., ([Nic, 2.1] [, 1 ] ): X = S 1 S b, 1-cell x, y, 2-cell u. u = xyx 1 y 1. π 1 (X) = x, y xyx 1 y 1, FOX ( 5.18) 2 : C 2 ( X Z[π1 (X)]F) C 1 ( X Z[π1 (X)]F); ũ x [x, y] [x, y] +ỹ x y = x(1 φ(y))+ỹ(φ(x) 1)., 5.18, 1 ( x) = φ(x) 1, 1 (ỹ) = φ(y) 1.,. (1 φ(y), φ(x) 1) (1 φ(x), 1 φ(y)) C : 0 C 2 C T 1 C 0 0. det(1 φ(x)) det(1 φ(y). det(1 φ(y) = 0. b 2 := u, b 1 = x, b 0 =. ( ) 1 φ(y) 1 [b 2 /c 2 ] = 1, [( b 2 )b 1 /c 1 ] = det = 1 φ(x), [( b 1 )b 0 /c 0 ] = det(1 φ(x)). (1 φ(x)) 0,, τ φ (C ) = n > 1, X = (S 1 ) n, F = Q(H 1 (X)) = Q(t 1,..., t n ), 1., X S 3 \ K (H = Z α )., Milnor Alexander : τ α (X) = K (t)/(1 t) Proof X. t 1 K (t), ( ) H ( X; Q(H)) ,., τ α (X) = det(tv T T )/(1 t)., K (t),., Mayer-Vietoris Gluing Y, X 1, X 2 X X = X 1 X 2 X 1 X 2 = Y. φ : Q(H 1 (X)) F, : j : Q(H 1 (Y )) Q(H 1 (X)), j k : Q(H 1 (X k )) Q(H 1 (X)) 26

27 C ( X) F q, C ( X k ) F q, C (Ỹ ) F q 17. τ φ (X) τ φ j (Y ) = τ φ j1 (X 1 ) τ φ j2 (X 2 ) up to ± φ(π 1 (X)) , X = S 1 S 2, solid torust 1, T 2. H 1 (T i ) = t i = Z, T 1 T 2 = T 1 = T 2 = S 1 S 1. T 1 T 2 T 1, T 1 T 2 T 2, t = t 1 = t 2. τ(s 1 S 2 ) = (t 1 1) 1 (t 2 1) 1 1 = (t 1) X = S 1 S n. Gluing, n > 1,. (t 1) 2 (n ) τ(s 1 S n ) = 1 (n ). Reidemeister, F,. ( ),., Reidemeister Spin c., ([Nic, 3.7] ) CW Milnor X CW. CW, (, ). H α : π 1 (X) H. Q[H] H ( ), Q(H). α. C : 0 C m (X) Z[π1 (X)]Q(H) C m 1 (X) Z[π1 (X)]Q(H) C 1 (X) Z[π1 (X)]Q(H) C 0 (X) 7.23., Milnor., Milnor. τ(c ) Q[H]/±H, Alexander. Alexander, TorH i (X; Q(H)) 0 i,,. A α (X) := i=0 ordh 2i 1 (X; Q(H)) ordh 2i (X; Q(H)) Q(H) A α (X) = τ α (X) Q(H)/ ± H. 17 Tureav refied torsion,. 27

28 7.25. R, K. R, : C : 0 C m C m 1 C 1 C 0 0 rank K H (C) = 0, Tor ( H (C) ) = H (C). C [c]. ζ R, m τ α (C, c) = ζ ord ( H j (C) ) ( 1) j+1 j=0 Proof.. [Mil4, Tu1, Nic]., Milnor,, X X, α., Milnor : τ α (X) = τ α (X ) 7.27., Whitehead, Reidemeister ([Tu1, 2 ] ). Whitehead Wh 1 (Z[G]) G., ; G, Whitehead. G = H (Bass, Heller, Swan). Wh 1 (Z[G]) = 0., G, Wh 1 (Z[G]) = 0. ( ) ([Tu1, ] ) L(p, q) L(p, q ). 1. l (Z/m), ±l 2 q = q (mod p). 2. q = ±q (mod p).,,. Hatcher 3 [], (Seifert ),. (2). 28

29 7.29 ( ). (p, q) = 1 p, q. L(p, q)., ζ Z/p, S 3 := {(z, w) C 2 z 2 + w 2 = 1 } (z, w) p,q ζ = (e 1/2pπ z, e 1q/2pπ w), Z/p S 3. L(p, q) S 3 /Z p. 3. L p, q S 3, π 1 (L(p, q)) = Z/p.,, H 1 H 2 Z/p., F Q(ζ), φ Z[Z/p] = Z[ζ] Q(ζ). t := φ(ζ)., Reidemeister, S 3. : E 1 j := Ej 0 := {(e j 1/2pπ, 0) S 3 } { (e 1θ, 0) S 3 2πj p θ 2π(j + 1) p { (z 1, se j 1/2pπ ) C 2 s R, z s 2 = 1 Ej 2 := { Ej 3 := (z 1, z 2 ) S 3 2πj p argz 2 2π(j + 1) p Z/p. Ej k k-ball, CW. c k j := Int(Ej k ),. ( ) : p 1 c 2 j = c 1 i, c 1 j = c 0 j+1 c 0 j, c 3 j = c 2 j+1 c 2 j., E k j i=0, ζ E k j = E k j+1, k {0, 1} ζ E k j = E k j+q, k {2, 3}., : p 1 c 2 j = ( ζ i )c 1 n, c 1 j = (ζ 1)c 0 j, c 3 j = (ζ r 1)c 2 j, i=0 rq 1 mod pz r.,. 0 Q(ζ) 1 ζa 1+ζ+ ζ Q(ζ) p 1 Q(ζ) 1 ζ Q(ζ) 0. ζ p 1, 0.., C φ. 7.7, τ φ (L(p, q)) = (1 t) 1 (1 t r ) 1. H 1 (L(p, q)) = Z/p s q = t, : τ φ (L(p, q)) = (1 s q ) 1 (1 s) 1. } }. } 29

30 L(p, q) L(p, q )., φ : π 1 (L(p, q)) Q(ζ) φ : π 1 (L(p, q )) Q(ζ), : u, k Z, t u (1 t r )(1 t) = (1 t kr )(1 t). u, : (1 t r )(1 t 1 )(1 t r )(1 t) = (1 t kr )(1 t k )(1 t kr )(1 t k )., {1, 1, r, r} = {k, k, kr, kr }., (Franz ). S := {j Z/pZ (j, p) = 1 }. Z {s j } j S Z : (1) j S a j = 0, a j = a j. (2) p ζ 1, j S (ζj 1) a j = 1., j S a j = , L, ( [ ] ).,. ( ) j S, m(j) := #{s {1, 1, r, r} s = j}, m (j) := #{s {1, 1, kr, kr} s = j}. a(j) m(j) m (j),. 0 = a(j) m(j) = m (j). (2).. r, r, k 8,., P (T ) C[T ]/(T p 1) : P (T ) = T u (T 1)(T r 1) (T k 1)(T kr 1) C[T ]/(T p 1)., k = 1 kr = t., T u+r (T 1)(T r 1) = (T k 1)(T r 1) C[T ]/(T p 1). (6),, (6) (ζ 1) 1 p 1 p (1 + 2ζ + + pζp 1 ) = 1 ( ζ i /p) p 1 (T u+r + 1)(1 ( ζ i /p) = 0,. k = 1 kr = t r = ±r.. q = ±q,., (1). i=0 i=0 30

31 (1)., q = ±l 2 q modp.. (p, k i ) = 1 k 1, k 2 Z, f k1,k 2 : S 3 S 3 f k1,k 2 (z 1, z 2 ) = ( z 1 1 k 1 z k 1 1, z 2 1 k 2 z k 2 2 )., f k k 1 k 2, : f k1,k 2 (ζ r,x (z 1, z 2 )) = ζ k1 r,k 2 x (z 1, z 2 ). [f k1,k 2 ] : S 3 / r,s S 3 / k1 r,k 2 s. p S 3, p B, k < p ζ k r,s B B =. U := p 1 k=0 ζk r,s B., q = l 2 q mod p. r 0 Z/p q. f llqr0 ( 1,q, l,lq ), : [f l,lq r 0 ] : L(p, q) L(p, q )., deg f l,lq r 0 l 2 q r 0 1 mod p.,. f l,lq r 0 H (S 3 ) H (S 3 ), Whitehead, f., f l,lq r 0. i 2 π i (L(p, q)) = π i (S 3 ), [f l,lq r 0 ] : π i (L(p, q)) π i (L(p, q )) [f l,lq r 0 ] : π i (S 3 ) π i (S 3 )., Whitehead, f. 8., Alexander Blanchfield. 8.1., R, : R R (involution) 18. M R Z- ψ : M M R,. a, b R, m, n M, ψ(am, bn) = abψ(m, n). ψ : M M R, ψ ψ, R- f : M M, ψ(a, b) = ψ (f(a), f(b)). ψ (Hermitian), ψ(a, b) = ψ(b, a). 18, ā = a,. = id R. 31

32 8.2. M R n a 1,..., a n. b ij := ψ(a i, a j ), n n B = {b ij } i,j n. n n B = {ψ (a i, a j )} i,j n., ψ ψ, : O, O T BO = B R = M = Z/m, = id R. A Z/m, ψ(x, y) = Axy. ψ (x, y) = A xy., ψ ψ, l (Z/m), A = l 2 A.,. 19.,,. 8.4 (Sylvester ). R R, = id R., ψ, m, n,. diag(1,..., 1, 1,..., 1, 0,..., 0) }{{}}{{} n m R = Z, indefinite 20 ψ,, (Serre ). R = Z, definite ψ,. M, R = Z/m, ψ. R (Q ), ψ ( [ ] ). R, t : R R ψ(ta, tb) = ψ(a, b), (ψ, t) isometry. isometry Milnor[Mil5]. 8.2 Linking. M n.,. P.D.: H k (N; Z) = H n k (N; Z).,, : H k (N; Z) H n k (N; Z) Z, (p, q) P.D. 1 (P.D.(p) P.D.(q)).. p k-cycle q (n k)-cycle,,., P, T P := {p P a Z ap = 0}. 19,, [MH, 1 ],.,,. 20 ψ R, Sylvester, 32

33 , Q/Z L N : T H l (N; Z) T H n l 1 (N; Z) Q/Z ( ). [x] T H l (N; Z) [y] T H n l 1 (N; Z) x C l (N; Z) y C n l 1 (N, Z), w C n l (N; Z) s Z w = sy. L N (x, y) := x, w /s Q/Z., N N, L N. 8.5., x, y, w, s.,. Proof. welldefined. 0 Z Q Q/Z 0 : B : H k+1 (N; Q/Z) H k (N; Z). B, H k (N; Z) H k (M; Q)., T H k (N; Z), L N (x, y) = B 1 (x) y,. : H k+1 (N; Q/Z) H k (N; Z) Q/Z., x B u, u, (u u ) y = 0., u u v H 2 (N; Q) v y = 0 ( y H k (N; Q) ). H k+1 (N; Q/Z) Poincaré H n k 1 (N; Q/Z) evaluation Hom(H n k 1 (N), Q/Z) B TorH k (N; Z) ˆλ Hom(TorH n k 1 (N; Q/Z) ˆλ x L N (x, ). ˆλ,,.,, L N (y, x)( 1) n(n k) L N (x, y).. 1, : 8.7. L(p, q) 3. T H 1 (L(p, q); Z) = H 1 (L(p, q); Z) = Z/p, θ.. L N (θ, θ) = q/p. Proof. S 1 = { z C z = 1 }, D 2 = { z C z 1 }, p, q, r, s pr qs = 1.. f : S 1 S 1 S 1 S 1, (z, w) (z q w p, z s w r )., U 1 := D 2 S 1, U 2 := S 1 D 2. U i S 1 S 1, x 1 x 2 def x 2 = f(x 1 ) x i U i. 33

34 ., U 1 U 2 / L(p, q) ( 21 ). H 1 (U 1 ) = Z H 1 (U 2 ) = Z µ K. : 0 Zµ p H 1 (U 1 ) H 1 (L(p, q)) 0., [K] H 1 (L(p, q)), pk U 1 = D 2 S 1 2-core( ) ( Mayer-Vietoris )., lk([k], [µ]) = 1/p., [µ] H 1 (L(p, q)) = Z/p, x Z/p, [K] = x[µ]. x. f f : H ( U 1 ) H ( U 1 ) ( ) = Z µ, λ, p r., : q s µ pµ + qλ, K rµ + sλ H 1 ( U 1 ). λ H 1 (U 1 ), [K] = s[µ]. s = q 1 mod pz,, q, : L N ( q 1 µ, µ) = 1/p H 2 (M; Z) = T H 2 (M; Z) 5 M,. T, b : T T Q/Z. b(x, y) = b(y, x), b(x, y) = b(y, x). [Wall], [Kawauchi-Kojima] b. 3, [?]. 3,., ( ) 8.9 ([Tu4]). M 3, β 1 (M) = 0. H := H 1 (M; Z)., : g, g H, τ(m) (g 1)(g 1) = λ M (g, g ) Σ H Q[H]/Z[H], Σ H h H h.,.,., r Z, H 1 (M; Z/r) H 3 [M] (M; Z/r) Z/r., 2 [M]. 3. λ (r)., α : π 1 (M) H 1 (M; Z) =: Γ. 22 α : H k (M; Z) H gr k (π 1(M)) H gr k (Γ). k = 3 [M] evaluate α ([M]) H gr 3 (Γ). 21 Dehn (p/q)

35 8.10 (Cochran-Gerges-Orr). M, M 3. ψ : H 1 (M; Z) H 1 (M ; Z) ψ(α ([M])) = α ([M ]), M M, r Z λ (r) (λ (r) ). 8.3 Blanchfield. Blanchfield [Bla]. [Hil, Kaw, FP]. X ( ), π 1 (X) Γ. Γ. X. Z[Γ] Γ, Q(Γ)., X X {t n } n Z, H k ( X; Z) H k (X; Z[Γ]) (, Shapiro )., Blanchfield, X (Z[Γ] ) : Bl : Tor Z[Γ] (H l ( X; Z)) Tor Z[Γ] (H n l 1 ( X, X; Z)) Q(Γ)/Z[Γ].. [x] Tor Z[Γ] (H l ( X; Z)) = Tor Z[Γ] (H l (X; Z[Γ])) [y] Tor Z[Γ] (H n l 1 ( X, X; Z)) = Tor Z[Γ] (H n l 1 (X, X; Z[Γ])), x C l (X; Z[Γ]), y C n l 1 (X, X; Z[Γ])., Z[Γ] w C n l (X, X; Z[Γ]), y = w. Bl([x], [y]) = g Γ x g, w / Q(Γ)/Z[Γ]., X, g Deck well-defined (x, y, ) Blanchfield (, [?, D.3]. ) Bl(y, x) = ( 1) (n l 1)l+1 Bl(x, y)..(, [Kaw, D.1]. ) 8.14 ( )., K : S 2k 1 S 2k+1 X = S 2k+1 \ ImK, f π 1 (X) Z =: Γ. H k ( X; Z) Z[t ±1 ]-. order k Z[t ±1 ] Alexander., 0 Z[t ±1 ] k - Z[t ±1 ] Z[t ±1 ]/ K 0 δ, H k+1 (X; Z[t ±1 ]/ K ) = H k (X; Z[t ±1 ])., : H k (X; Z[t ±1 ]) δ 1 H k+1 (X; Z[t ±1 Poincaré duality ]/ K ) H k (X, X; Z[t ±1 ]/ K ) 35

36 H k (X; Z[t ±1 ]/ K ) Hom(H k (X; Z[t ±1 ]), Z[t ±1 ]/ K ). (Ext 0 = 0 ),, (X, ) (X, X)., ( ) Blanchfield, : Bl K : H k (X ; Z) 2 = H k (X; Z[t ±1 ]) 2 Z[t ±1 ]/ K. Bl K (non-singular), Hermitian., K = K. Bl K, ( [Hil, 3] )..,. k 2, 2k 1 3 K 1 K 2 (, π 2 (S 2k 1 \ K j ) = = π k 1 (S 2k 1 \ K j ) = 0)., K 1 K 2, Bl K1 Bl K2 [2]. k 2, Witt 23, Z ( Z/2) ( Z/4). Blanchfield ([3] ). Seifert, Bl K F S 3 K, V Mat(2g 2g; Z). {e i } i 2g H 1 (S 3 \ F ; Z) ( 6.1 ). Λ := Z[t ±1 ]., Λ 2g H 1 (S 3 \ K; Λ); (p 1,..., p 2g ) e i p i 1 i 2g Φ : Λ 2g /(tv V T )Λ 2g = H1 (S 3 \ K; Λ) (7), : Λ 2g /(tv V T )Λ 2g Λ 2g /(tv V T (v,w) v(t 1)(V )Λ 2g tv T ) 1 w Q(t)/Λ Φ Φ H 1 (S 3 \ K; Λ) H 1 (S 3 \ K; Λ) (v,w) Bl K (v,w) = Q(t)/Λ [FP], (, [Hil, 2] ). Bl K, (7).,. t, ω C w = 1. (1 ω)v (1 ω)v T, L ω-, σ ω (K). 23 Witt,, Grothendieck.. 36

37 [Lic, 8 ]., 8.17 ([Lic, 8.19]). K. ω, σ ω (K) = 0., Witt W (Z[t ±1 ], ϵ). ([Hil] ),. Blanchfield. Milnor [Mil1] ( [ ] ).,,. Blanchfiled, 3 [1]. 9 CW. 9.1 (CW ). : D n = {x R n x 2 1}, D n = {x R n x 2 = 1} = S n 1. X CW -, = X 1 X 0 X 1 X = n 0 X n (, X ). 1. X 0,. 2. X n n-skeleton I n, α I n φ α : D n = S n 1 X n, X n+1 : X n+1 = X n D n = X n D n /( D n x φ α In φα α (x)). α I n α I n, CW. CW-, ( ),. CW. X x X. π 1 (X) := {(S 1, ) (X, x) f( ) = x }/(rel. ) X π 1 (X). G = g 1,..., g n r 1,..., r m CW Xs.t. π 1 (X) = G. x 0, X 1 S 1 n S 1 S 1., 37

38 r i = g ϵ 1 i 1 g ϵ k ik, k a 0 a k, a j 1 a j j S 1 (, ϵ j = 1, ϵ j = 1, ). X, CW.,, π 1 (X) = G., X CW., X 0., X, S 1 n S 1 S 1., 2-cell, k a 0 a k, 1-cell., π 1 (X) = G. : : 9.2 ( ). C X p : C X x X x U p 1 (U) C p U. C Deck ( ) g : C C p g = p. Deck,. p (regular), p : π 1 (C) π 1 (X), π 1 (X)., x X, p 1 (x). π 1 ( X) = 1 X p : X X X CW. π 1 (X) H, p : Y X, p (π 1 (Y )) = H. X, π 1 (X), X CW., X. X, h : X X, p h = p., p : C X π 1 (X)/p (π 1 (C))., C = X, G = π 1 (X). [Adams] C. Adams, The Knot Book. An elementary introduction to the mathematical theory of knots. W. H. Freeman and Company, New York, 1994 (, 1998). [Bla] R. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. 65: (1957) [Bro] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, [CF] R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and Co. 1963, or Grad. Texts Math. 57, Springer-Verlag, 1977 (, 1967). [CFH] A. Conway, S. Friedl, G. Herrmann, Linking forms revisited, preprint, arxiv.org math arxiv: [FP] Stefan Friedl and Mark Powell. A calculation of Blanchfield pairings of 3-manifolds and knots arxiv:

39 [FV] S. Friedl and S. Vidussi. A survey of twisted Alexander polynomials. Preprint 2010, 42 pages. arxiv: [Gor] Gordon, Some aspects of classical knot theory, Lecture Notes in Mathematics 685, Springer-Verlag 1 60 [Hat] A. Hatcher, Algebraic topology, Cambridge University Press (2002). [Hil] J. Hillman, Algebraic Invariants of Links, Series on Knots and Everything, 2-nd edition. [ ], ( ) [ ] [ ], Reidemeister torsion (How to use the Reidemeister torsion), theset.las.osakasandai.ac.jp/fledglings/notes/kadokami.pdf [Kaw] A. Kawauchi (ed), A Survey of Knot Theory, Birkhaüser Verlag, Basel, (, ) [ ],,, [1] A. Kawauchi, Three dualities on the integral homology of infinite cyclic coverings of manifolds, Osaka J. Math. 23 (1986), [2] C. Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975), [3] J.P. Levine, Algebrai structure of knot modules, Lecture Notes in Mathematics 772. Springer-Verlag [Lic] W. B. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, 175. Springer-Verlag, New York, [Lin] X.S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), [Lyn] R. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52, (1950) [Mas] G. MASSUYEAU AN INTRODUCTION TO THE ABELIAN REIDEMEISTER TORSION OF THREE- DIMENSIONAL MANIFOLDS, arxiv: v2 [math.gt] 12 Oct [McC] J. McCleary, A user s guide to spectral sequences, Second edition. Cambridge Studies in Advanced Mathematics, 58. Cambridge University Press, Cambridge, [Mil1] J. Milnor, Infinite cyclic coverings, [Mil2], A duality theorem for Reidemeister torsion, Ann. of Math. (2), 76, 1962, [Mil3], Two complexes which are homeomorphic but combinatorially dis- tinct, Ann. of Math. (2), 74, 1961, [Mil4], Whitehead torsion, Bull. Amer. Math. Soc., 72, 1966, [Mil5], On Isometries of Inner Product Spaces. [MH] J. Milnor, Husemoller, Symmetric Bilinear Forms [ ],,, [ ] ; ;, Alexander, 5 [Ni] Yi Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), no. 3, [Nic] L. I. Nicolaescu. The Reidemeister torsion of 3-manifolds, volume 30 of de Gruyter Studies inmathematics.walter de Gruyter & Co., Berlin, [NY] F. Nagasato and Y. Yamaguchi: On the geometry of the slice of trace-free SL2(C)-characters of a knot group, Math. Ann. 354 (2012), [OS] Peter S. Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), , [OSJS],,,, Alexander,. [Rei] Kurt Reidemeister. Homotopieringe und Linsenräume. Abh. Math. Semin. Hamb. Univ., 11: , [Rol] D. Rolfsen, Knots and Links, AMS Chelsea Publishing, [ ],,, [ ],, 21, 22 [Tro] H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. 76 (1962), [Tu1] Turaev V., Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Notes taken by Felix Schlenk, Birkhäuser Verlag, Basel, [Tu2] Turaev V., Torsions of 3-dimensional Manifolds, Springer [Tu3] V. Turaev. Reidemeister torsion in knot theory. Uspekhi Mat. Nauk, 41(1):97-147, 240, 1986 (in Russian). English translation: Russian Math. Surveys, 41: , [Tu4] V. Turaev. Torsio invariants of Spin c -structures on 3-manifolds. Math. Res. Lett., 4(5): (1997). [Yj] Yajima, On a characterization of knot groups of some knots in R 4, OJM 39