Transcription

1

2

3 Kauffman Lambropoulou [50 53]( ) ( ) tangle (2-) Conway [0] Conway 0 ( ) ( ) Conway 2 4 (+?) ( ) Conway Kauffman Lambropoulou Kauffman Krebes Schubert DNA

4 8-207

5 . Conway Kauffman Krebes DNA 0 Appendix A. Jones Fox 23 Appendix B. 36 Appendix C. Tait

6

7 T. Conway Conway [20, 03] - : Conway Conway (square dance) A, B, C, D A D B C 2 A D B C [ ] [ ] [ ] C D D C C D D C T T [ ] 4 90 T Conway 7

8 T T R ( 2 ) TTRTR - 2- t T R 3 t (R T) 3 (t) = t. ( ) T T T R T T R T T T R T T Conway Kauffman Lambropoulou [52; Section 6] 2 SL(2, Z) SL(2, Z) := { ( ) } ( a b p a, b, c, d Z, ad bc = v = c d q) Z 2 {0} Q { } [v] := p q 8

9 (q = 0 p ( a q = ) A = c [Av] = ap + bq cp + dq R, T SL(2, Z) ( ) ( ) 0 R =, T = 0 0 ) b SL(2, Z) d R 90 - T 2 = S S ( S D 2 ) R(z, w) = ( w, z), T (z, w) = (zw, w), (z, w) S S ( ) S R T p q Q { } p, q (q = 0 p = ) ( ( p 0 A = q) ) A SL(2, Z) v Z 2 {0} [Rv] =, [T v] = [v] + [v] Q { } R, T R(x) =, T (x) = x + (x Q { }) x SL(2, Z) R T A R, R, T, T A = X X k (X i {R, R, T, T }) [ ( ) ] [ ( ) ] ([ 0 ( ) ]) p p 0 = = A = (X q X k ) q R = R 3, T = RT RT R Conway. () SL(2, Z) ( ) ( ) 0 (2) SL(2, Z) R =, T = 0 0 9

10 ( ) ( ) ( ) ( ) 0 ± q a c (2) a, c, q, r Z a = qc + r = ( 0 0 ±c r ) A ( ) SL(2, Z) R, R, T, T 0 Conway [20, 03] 2 (SQD) T (SQD2) x x R 0 Q (SQD2) 0 ( ) 0-2 TTRTR 0 T T 2 R 2 T 2 R 2 TT ( ).. R 0 R T 0 x := p q (p, q N, p q) r T x := q p q 0

11 p = q x = 0 p < q x x R x 2 := q q p T x 3 := r (r N, r q p) q p ( 0 T ) x x 3 x 3 T 0. n Conway n, 2, 3-2 : ( ) m m S = { (x, y) R 2 x 2 + y 2 = } R 3 m- (link) m (component) - (knot) L #L 3 4 (Hopf link) (Borromean rings) (polygonal link)

12 L L i ( ) (oriented link) L i f i (S ) = L i f i : S R 3 2 K K 2 R 3 ( ) R 3 L m- L AB C ABC L AB L AB AC CB m- L L L L L C C B B A A m- L, L 2 (equivalent) (ambient isotopic) L L 2 m- L, L 2 L R 3 π : R 3 R 2, π(x, y, z) = (y, z) (LD) L π R 2 (LD2) L π (LD3) L π 2 2

13 (LD4) L π ( π(l) 3 ) L R 3 L L (diagram) (LD),..., (LD4) L D = π(l ) z- (link diagram) L L ( ) (oriented link diagram) ( ) R 3 ( ) - 3 : Reidemeister 3 R 3 π : R 3 R 2, π(x, y, z) = (y, z) (LD),..., (LD4) 930 Reidemeister (RI) (RII) (RIII) 3

14 ( ) (RI) (RII) (RIII) (RI),(RII),(RIII) Reidemeister (Reidemeister move) Reidemeister ( ) (i) R 2 (ii) (iii) R 2 ABC ( ABC) D = AB MN ( M AB N BC AC A, B, C ) D AB AC BC N M 4

15 A B M N C A B M N C D ( ) D ( ) D ( ) 2 D, D - 3 (Reidemeister [86, 87]) L, L R 3 ( ) D, D L, L ( ) L L D D Reidemeister (RI),(RII),(RIII) - 4 : B n, S n n (standard ball) n (standard sphere) B n = { (x,..., x n ) R n x x 2 n }, S n = { (x,..., x n, x n+ ) R n+ x x 2 n + x 2 n+ = }. B 2 D 2 (2 ) (disk) B n n ( ) S n n ( ) S n S n ± Sn 0 S n + = { (x,..., x n, x n+ ) S n x n+ > 0 }, S n = { (x,..., x n, x n+ ) S n x n+ < 0 }, S n 0 = { (x,..., x n, x n+ ) S n x n+ = 0 }. B 3 ϕ : B 3 B B = ϕ(s 2 ) S 2 +, S 2, S B +, B, B B +, B, B 0 B n n- (tangle) 3 B B t (B, t) 2 ( 5

16 [50] ) (Tang) t B, t B = t. (Tang2) ( t B ± ) = n, t S 2 0 =. 3 B t B 3 B 2 n- (B, s), (B, t) (equivalent) s = t (B, s) (B, t) (B, s) (B, t) s t n- (B, t) (trivial) 2 (= 2 ) B t (TT) B = B 0 2 (TT2) t t = ( ) t, B0 (TT3) t t i ( t i B ± ) =. B B 3 B (TT) 2 q : B (B, B +, B, ) = (B 3, S 2 +, S 2, {0} D 2 ) (x, y, z) (y, z) n- (B, t) (regular position) (RP) t B ± B ± n (RP2) q(t) n- (B, t) = {0} D 2 (= + ) ((Tang) t {0} S ) n- (tangle diagram) n- Reidemeister (RI),(RII),(RIII) 6

17 Reidemeister (RI),(RII),(RIII) 2-4 (Reidemeister) t, t 3 B t = t n- D, D 2 t, t t t D D Reidemeister (RI),(RII),(RIII) 7

18 2. [35,47,63,92,02] [60] 2 - : a,..., a n ( ) (continued fraction) (2.) a + a 2 + a a n + a n (2.2) [a, a 2,..., a n ] a,..., a n [a k, a k+,..., a n ] = a k + a k a n + a n 0 a R (2.3) a + = = + a, = 0, 0 =. [a,..., a n, a n ] a, a 2,..., a n R a n R { } 2- a, b R [a] = a, [a, 0] = a + 0 = a + =, [a, ] = a + = a + 0 = a, [a 0, 0, a 2 ] = a a 2 = a 0 + a 2 = [a 0 + a 2 ] 8

19 [a, b, 0] = a + b + 0 [a, b, ] = a + b + = a + b + = a + = a + 0 = a, = a + b + 0 = a + b = ab + (b 0), b (b = 0) a,..., a n R, a n R { } i {2,..., n} (2.4) [a,..., a n ] = [a,..., a i, [a i,..., a n ]] a,..., a n 0 a i = 0 (i {,..., n }) (2.5) [a, a 2,..., a n ] = [a, a 2,... a i 2, [a i, 0, a i+,..., a n ]] = [a, a 2,... a i 2, [a i + a i+, a i+2,..., a n ]] = [a, a 2,..., a i 2, a i + a i+, a i+2,..., a n ] 2-2 () α R n N (2.6) α = [a, a 2,..., a n, α n ] (a, a 2,..., a n Z, α n R) α (n ) (2.6) (i) a 2,..., a n N. (ii) α n > ( n 2 ). α (n ) (2) α R n N α α = [a, a 2,..., a n ] (a Z, a 2,..., a n, a n N, n 2 a n > ) α ( ) (3) α Q n N α (2.6) α n Z α 2-3 () 0 7 = = (2) ω := ( ) = = [, 2, 3] 0 7 ( ) (n ) ω = [,,...,, α] 9 = [, 2, 3] 0 7 ( α = 2 5 )

20 ω = + ω = [, α] 5 2 α = ω = + α 2 = 2( 5 + ) 5 = + = α ω = + + α ω ω 2 ω = [,, α] ω = α ω (n ) ω = [,,...,, α] (2) Appendix B. 3 (n ) 2-2 : 2-4 α R α Q α ( ) = = α I. α α = [α] II. α α = m n (m Z, n N) m = q n + r (0 r < n) 20

21 q, r α r 0 0 < r < n n q, q 2,..., q n, q n+, r, r 2,..., r n n = q 2 r + r 2, (0 < r 2 < r ), r = q 3 r 2 + r 3, (0 < r 3 < r 2 ), r 2 = q 4 r 3 + r 4, (0 < r 4 < r 3 ),.. r n 2 = q n r n + r n, (0 < r n < r n ), r n = q n+ r n n, r, r 2,..., r n q 2, q 3,..., q n, q n+ m n = q + r n, n = q 2 + r 2, r r r = q 3 + r 3, r 2 r 2 r 2 = q 4 + r 4, r 3 r 3.. r n 2 = q n + r n r n = q n+ r n r n r n, r n 2 = q n + = [q n, q n+ ], r n q n+ r n 3 = q n + r n 2 r n 2 r n. n r = q 2 + r 2 r. = q 2 + = q n + [q n, q n+ ] = [q n, q n, q n+ ] [q 3,..., q n, q n+ ] = [q 2, q 3,..., q n, q n+ ] 2

22 α = m n = q + r n = q + [q 2, q 3,..., q n, q n+ ] = [q, q,..., q n, q n+ ] q n+ r n r n q n+ > α = [q, q 2,..., q n, q n+ ] α 2-3 : 2-5 m, n N a, b Z, a i, b j N (i = 2,..., m, j = 2,..., n ) α m, β n [a,..., a m, α m ] = [b,..., b n, β n ] m n a i = b i (i =,..., m ), 2 α m = [b m, b m+,..., b n, β n ] m = n a i = b i (i =,..., m ), α m = β m ( ) I. m = 2 α = [α ] = [b,..., b n, β n ] 2 II. m 2 i =,..., m s i, t i R s i = [a i+,..., a m, α m ] t i = [b i+,......, b n, β n ] s i >, t i > ) i m 2 s i > a i+, t i > b i+ i = m s i = α m > m < n t i > b m, m = n t i = β n > a + s = b + t a, b Z 0 < s, t < a = b, s = t 22

23 ) a, b α, β 0 < α, β < a + α = b + β a = b α = β a b = β α < a b < a, b Z a b = 0 a = b α = β a = b, s = t s = a 2 +, t = b 2 + a 2 + = b 2 + s 2 t 2 s 2 t 2 a 2 = b 2, s 2 = t 2 a i = b i, s i = t i (i =,..., m ) m = n s m = α m, t m = β m α m = β m m < n s m = α m, t m = [b m,..., b n, β n ] α m = [b m,..., b n, β n ] α α = [a, a 2,..., a n ] (a Z, a 2,..., a n, a n N) m, n N a, b Z, a i, b j N (i = 2,..., m, m, j = 2,..., n, n) [a,..., a m, a m ] = [b,..., b n, b n ] m n a i = b i (i =,..., m ), 2 m = n a m = b m m + = n b m = a m, b m+ = 2-7 n N b Z, b j N (j = 2,..., n, n) [b,..., b n, b n ] n = n = 2 b 2 = ( ) n 2 [b,..., b n, b n ] n = 2 b 2 = 23

24 n = 2 [b, b 2 ] = b + b 2 [b, b 2 ] b 2 b 2 = n 3 [b,..., b n ] = b + [b 2,..., b n ] [b 2,..., b n ] b j N (j = 2,..., n, n) n 3 [b 2,..., b n ] > > [b 2,..., b n ] (> 0) [b 2,..., b n ] n 3 [b,..., b n ] ( 2-6 ) I. m = [a ] = [b, b 2,..., b n, b n ] [b, b 2,..., b n, b n ] 2-7 n = a = b n = 2 b = a, b 2 = II. m 2 i =,..., m s i, t i R s i = [a i+,..., a m, a m ] t i = [b i+,......, b n, b n ] i =,..., m s i >, t i > 2-5 a i = b i, s i = t i (i =,..., m 2) s m 2 = a m + s m = a m + a m, t m 2 = b m + t m (i) a m = t m a m + a m = b m + t m t m = a m = b m t m = [b m, b m+,..., b n ] 2-7 m = n n = m + b n = m = n = t m = b m a m = b m n = m + b n = = t m = b m + b m = 0 b m N (ii) a m 2 t m > a m t m 2-5 a m = b m, a m = t m t m = [b m,..., b n, b n ] 2-7 m = n n = m + b n = m = n a m = t m = b m 24

25 n = m + b n = a m = t m = [b m, ] = b m [2, 3, 2] = = = 6 7, [2, 3,, ] = = [2, 3, 2] = [2, 3,, ] + = = 6 7 [a,..., a n ] n n α () α = [a,..., a n ] n N a Z, a 2,..., a n N (2) α = [a,..., a n ] n N a Z, a 2,..., a n N ( ) () α 2-4 α = [a,..., a n ] (a n > ) n N a Z, a 2,..., a n N n α = [a,..., a n ] α n α = [a,..., a n, ] α 2-6 (2) 2-0 a,..., a n, a n Z α = [a,..., a n ] Q { } () α ± = [a ±, a 2,..., a n ]. (2) /α = [0, a, a 2,..., a n ]. (3) α = [ a, a 2,..., a n ]. (4) i {, 2,..., n} a i a i = b i + c i (b i, c i Z) α = [a,..., a i, b i, 0, c i, a i+,..., a n ]. 25

26 (5) a n > α = [a,..., a n, a n, ], a n = α = [a,..., a n + ], a n < α = [a,..., a n, a n +, ], a n = α = [a,..., a n ]. ( ) (), (2), (3) (4) β = [c i, a i+,..., a n ], γ = [a i+,..., a n ] b i + β = b i + [c i, a i+,..., a n ] = b i + c i + γ = a i + γ = [a i, a i+,..., a n ] [a,..., a i, b i + β] = [a,..., a i, [a i, a i+,..., a n ]] = [a,..., a i, a i, a i+,..., a n ] = α b i + β = b i + [c i, a i+,..., a n ] = [b i, 0, c i, a i+,..., a n ] [a,..., a i, b i +β] = [a,..., a i, [b i, 0, c i, a i+,..., a n ]] = [a,..., a i, b i, 0, c i, a i+,..., a n ] (5) a n > a n = (a n ) + = (a n ) + = [a n, ] α = [a,..., a n, a n, ] a n = [a n, a n ] = a n + a n = a n + α = [a,..., a n + ] a n < a n = (a n + ) = (a n + ) + = [a n +, ] α = [a,..., a n, a n +, ] a n = [a n, a n ] = a n + a n = a n α = [a,..., a n ]. α α = [a, a 2,..., a n ] (a, a 2,..., a n Z) a 2,..., a n 0 m < n m N b Z, b 2,..., b m Z {0} α = [b, b 2,..., b m ] 26

27 2-4 : 2- [a, a 2,..., a n ] (a,..., a n Z) (canonical form) (i) a 0 i =, 2,..., n a i > 0 i =, 2,..., n a i < 0 a = 0 i = 2,..., n a i > 0 i = 2,..., n a i < 0 (ii) n 2-2 ( ) 2-0(5) α (i) 2-4 α > 0 α = [a, a 2,..., a n ] (a N {0}, a 2,..., a n N, a n > ) (i) α < 0 α > 0 α = [a, a 2,..., a n ] (a N {0}, a 2,..., a n N, a n > ). α = [a, a 2,..., a n ] = [ a, a 2,..., a n ] (i) α α = [a,..., a m ] = [b,..., b n ] 2 m n I. m = [a ] = [b,..., b n ] [b,..., b n ] n = a = [a ] = [b ] = b II. m 2 α a 2,..., a m, b 2,..., b n (a 0, b 0 a, b ) α > 0 i =,..., m s i := [a i+,..., a m ], t i := [b i+,..., b n ] 27

28 i =,..., m 2 s i = a i+ + s i+ > a i+, t i = b i+ + t i+ > b i+ a + s = b + t a, b Z 0 < s, t < 2-5 a = b, s = t [a 2,..., a m ] = s = t = [b 2,..., b n ] a i = b i, s i = t i (i =,..., m 2) m < n a m > s m = a m > t m = b m + [b m+,..., b n ] > b m a m = b m, s m = t m a m = s m = t m = [b m,..., b n ] m = n a m = t m 2 = s m 2 = a m + a m = a m + t m 2 = [b m,..., b n ] n = m m, n n = m m < n m = n a m + = [a m, a m ] = s m 2 = t m 2 = [b m, b m ] = b m + a m b m a m > b m > a m b m 2-5 a m = b m, a m = b m a m = b m = = a m a m = b m b m α > 0 α α < 0 i =,..., m s i := [ a i+,..., a m ], t i := [ b i+,..., b n ] a i = b i, s i = t i (i =,..., m 2) m < n a m + = [ a m, a m ] = s m 2 = t m 2 = [ b m, b m ] = b m + a m b m a m > b m > a m b m 2-5 a m = b m, a m = b m a m = b m = = a m a m = b m b m 28

29 α < 0 α 29

30 3. [52] 3 - : 3 B 3 2- t t = t B 3 4 ( 0,, ), (0,, ) ( (, 0,, ), 0,, ) ϕ : (B 3, t) (B, ϕ(t)) 2- (B, ϕ(t)) 3-2- (B, t) (rational tangle) 2- (B, t 0 ) h : (B, t) (B, t 0 ) 3-2 () 2- [0], [ ] (2) n Z [n], 2- [n] [n] [n] = [ ] [0] [ ] = [0] n n n (n<0) n (n>0) 30

31 n n n (n<0) n (n>0) (3) : +,,, ( ) in, ( ) rot 2- {0} D 2 () ( ) s, t 2- ( ) s + t s + t := s t (2) 2 2- ( ) s, t 2- ( ) s t s t := s t 3

32 (3) 2- ( ) t t 2- ( ) t t (mirror image) (4) ( ) rot 2- ( ) t 2- ( ) t rot, t rot t (5) ( ) in 2- ( ) t 2- ( ) t in, t in t t in := -t, t in := -t -t -t t ±in t (inversion) 3-3 () 2- t t in = (t rot ) = ( t) rot, (t in ) in t (t in ) in, (t rot ) rot t (t rot ) rot. (2) 2- s, t (s + t) = ( s) + ( t), (s t) = ( s) ( t). (3) n Z { } [n] in = ( ) in [n], = [n]. [0] in = [ ], [ ] in = [0]. [n] 3-3 : 3-4 (twist form) [0] [ ] [±] 32

33 (i) [0] ( [s k ] + + ( ( [r 3 ] [s ] + [r ] [s 0] ) ) + [s 2 ] ) ) + + [s k+ ] [r 2 ] [r 4 ] (ii) [ ] ( ( ( [r k ] [s 3 ] + [r ] [s ] + [r 0 ] + [s 2] s i, r i Z 0 ) ) ) ) + [s 4 ] [r 2 ] [r k+ ] (i) [s 0 ] = [ ] (ii) 3-5 ( ) Cromwell [; Chapter 8] S 2 4 ( NW 0,, 2 ), SW (0,, ) (, NE 0, , ( ), SE 0, 2, ) H, V 4 [; p.20]( [9; Chapter 2] ) fixed fixed H V S 2 4 NW, SW, NE, SE f S 2 B 3 f 2- t f 2- f(t) S 2 4 NW, SW, NE, SE f, f 4 f, f f(t) f (t) ( ) 2- t 4 NW, SW, NE, SE B 3 h [0] ( [ ]) f := h S 2 H, V 4 {NW, SW, NE, SE} t [0] H, V 33

34 S 2 B 3 H ± B 3 2- s s + [±] V ± B 3 2- s s [±] t [0] [±] 2-2- (B 3, t) 2- (B 3, t) () 2- (2) T ( T = [s k ] + + ( [r 3 ] ( [s ] + ( [r ] [s 0] [r 2 ] ) + [s 2 ] ) ) ) + + [s k+ ] [r 4 ] [s 0 ] = [ ] T 2- [s 0 ] i {, 3,..., k} 0 s i s i, 0 s i+ s i+ s i, s i+ ( ( ( [s i] + [r i ] + [r 3 ] [s ] + ( [r ] [s 0] [r 2 ] ) ) + [s 2 ] ) + [r 4 ] 2- t (truncation) ( ) () ) + [s [r i+ ] i+] 0 [0] [ ] [0], [ ] 2- D 2 2 [0] [ ] [0], [ ] 2- t n T T p t (n ) 2- T [±] + T, T + [±], [±] T, T [±]. T 2- u t 2- u T 2- U U p U T 2- U U p U p 2- U [±] + U, U + [±], [±] U, U [±]. 34

35 T flype T T U T 2- U U () (2) T 0 () 2- T (n ) S 2- S T n T p T (n ) 2- T [±] + T, T + [±], [±] T, T [±]. T 2- T U T 2- U p U T 2- U T U T U p U p 2- U [±] + U, U + [±], [±] U, U [±]. U T U T T = [±] + T U = [±] + U U U T T U T (2) 3-4 : [±] + T [±] T 2- (flype) T flype T (rational) 35

36 t ( ) t ( ) 2- ( ) t hflip t (horizontal flip) t ( ) 2- t vflip t (vertical flip) t hflip t t vflip 2-2- T (3.) [±] + T T hflip + [±], [±] T T vflip [±]. 2- ( ) s, t (3.2) (s + t) hflip = s hflip + t hflip, (s t) hflip = t hflip s hflip, (s + t) vflip = t vflip + s vflip, (s t) vflip = s vflip t vflip ( ) t () t hflip t t vflip. (2) (t in ) in = (t rot ) rot t. (3) t in t in, t rot t rot. 36

37 T ( ) t T () [0], [ ], [±] () n n () T n B 2 T n T T + [±], [±] + T, T [±], [±] T 3-6 T T (T ) hflip (T ) vflip T ± := T + [±] (T ± ) hflip T ± (T ± ) vflip ( + ) (T ± ) hflip = T = T ± by induction T (T ± ) vflip = flype T by induction (hflip & vflip) T = T ± 2 T 2± := [±] + T (T 2± ) hflip T 2± (T 2± ) vflip 3 T 3± := T [±] (T 3± ) hflip T 3± (T 3± ) vflip (T 3± ) hflip = T flype T by induction (hflip & vflip) T = T 3± (T 3± ) vflip = T T = T 3± by induction (vflip) 4 T 4± := [±] T (T 4± ) hflip T 4± (T 4± ) vflip 3 (2) (T in ) in = T (T in ) in = (T hflip ) vflip () (T in ) in = (T hflip ) vflip T hflip T 37

38 (T in ) in = (T rot ) rot 3-3 (3) (2) 3-3 T in ( (T in ) in) in T in T rot T rot 3-0 m, n Z t [m] + t + [n] t + [m + n], [m] t [n] t [m + n]. ( ) (3.) 3-9 t [±] + t t + [±], [±] t t [±] [m] + t + [n] = ([m] + t) + [n] (t + [m]) + [n] = t + ([m] + [n]) = t + [m + n] [m] t [n] t [m + n] 3-5 : 3- (standard form) [0] [ ] [±] ( (( ) ) [a n ] + [a n 2 ] [a n ] [a 2 ] ) + [a ] a 2,..., a n Z {0} [a ] [0] [a n ] [ ] n t + = t t * = t 38

39 3 2 4 = ( [3] ) + [4] [ 2] 3-2 ( ) t ( t = [s k ] + + ( [r 3 ] [s ] + ( [r ] [s 0] [r 2 ] ) ) + [s 2 ] ) ) + + [s k+ ] [r 4 ] s 0, s i, r i Z (i =,..., k + ) [s 0 ] = [ ] 3-0 t ( ((( ) ) ) ) t [s 0 ] + [s + s 2 ] + + [s k + s k+ ] [r + r 2 ] [r 3 + r 4 ] s i + s i+ = 0 r i + r i+ = 0 t t in t t 3-9(3) 3-3 t n ( ) t [n] [n] +, t 3-0 n 0 [n] t. t + [n] [n] + t = n t t n = t hflip [n] 3-9() t hflip [n] t [n] 39

40 n < : 3-4 (continued fraction form) [[a ], [a 2 ],..., [a n ]] := [a ] + [a 2 ] + [a 3 ] + a, a 2,..., a n Z... + [a n ] + [a n ] a 2,..., a n 0 [[a ], [a 2 ],..., [a n ]] = [[b ], [b 2 ],..., [b m ]] (b 2,..., b m Z {0}) 3-5 ( ) t ( (( ) ) t = [a n ] + [a n 2 ] ) + [a ]. [a n ] [a 2 ] a 2,..., a n Z {0} [a ] [0] [a n ] [ ] n ( [a n ] = [ ] n ) n [[a ], [a 2 ],..., [a n ]] [a n ] = [ ] [[a ], [a 2 ],..., [a n ]] = [[a ], [a 2 ],..., [a n ]] n = t = [a ] ( n = t = [a 3 ] ) ( ) + [a ] = [a ] + [a 2 ] [a 2 ] [a 3] [a 3 ] = [ ] t = [a ] + [a 2 ] [a 3] [ ] 3-3 t = [a ] + [a 2 ] + [a 3 ] n 5 n 2 t := (( ) ) [a n ] + [a n 2 ] + [a 3 ] [a n ] 40

41 ( t = t ) + [a ] [a 2 ] ( t = t ) + [a ] [a 2 ] [a 2 ] + + [a ] [a ] + t [a 2 ] + t t [a n ] = [ ] [[a 3 ],..., [a n ]] t [[a ], [a 2 ], [a 3 ],..., [a n ]] [a n ] [ ] [[a 3 ],..., [a n ]] t [[a ], [a 2 ], [a 3 ],..., [a n ]] n ( [3] ) + [4] = [[4], [ 2], [3]] = [4] + [ 2] [ 2] + [3] 3-6 a,..., a n Z T = [[a ], [a 2 ],..., [a n ]] () T + [±] = [[a ± ], [a 2 ],..., [a n ]]. (2) T = [[0], [a ], [a 2 ],..., [a n ]]. (3) T = [[ a ], [ a 2 ],..., [ a n ]]. (4) a i (i {, 2,..., n}) a i = b i + c i (b i, c i Z) ( ) T = [[a ], [a 2 ],..., [a i ], [b i ], [0], [c i ], [a i+ ],..., [a n ]]. () T = [[a 2 ],..., [a n ]] T = [a ] + T T + [±] = [a ] + T + [±] = [a ] + [±] + T ( 3-0) = [a ± ] + T = [[a ± ], [a 2 ],..., [a n ]] (2) T = [0] + T = [[0], [a ], [a 2 ],..., [a n ]]. (3) n Z (3.3) [n] = [ n] 4

42 T = [a ] + = [ a ] + [a 2 ] + [ a 2 ] + [a 3 ] + [ a 3 ] + = [[ a ], [ a 2 ],..., [ a n ]] [a n ] + [a n ] [ a n ] + [ a n ] (4) S := [[c i ], [a i+ ],..., [a n ]] () [b i ] + S = S + [b i ] = [[b i + c i ], [a i+ ],..., [a n ]] = [[a i ], [a i+ ],..., [a n ]] [[a ], [a 2 ],..., [b i ] + S] = [ [a ], [a 2 ],..., [a i ], [[a i ], [a i+ ],..., [a n ]] ] = T [[b i ], [0], [c i ], [a i+ ],..., [a n ]] = [[b i ], [0], S] = [b i ] + [0] + S = [b i ] + S [[a ], [a 2 ],..., [a i ], [b i ], [0], [c i ], [a i+ ],..., [a n ]] = [[a ], [a 2 ],..., [a i ], [b i ] + S] = T 3-7 T = [[a ], [a 2 ],..., [a n ]] n (3.4) a a2 a3 an- an 42

43 n (3.5) a a3 an- a2 an n = n n (n 0), (n < 0) ( ) n n = n 3 n 2 (3.4) T = [[a 3 ],..., [a n ]] 3-3 T = [[a ], [a 2 ],..., [a n ]] = [a ] + [a 2 ] + = [a ] + T T ( T ) [a 2 ] a3 an- an a a2 (3.4) n 3-8 [0] [ ] [[a ], [a 2 ],..., [a n ]] 43

44 (i) a i (i =, 2,..., n) (ii) a i (i =, 2,..., n) (iii) a = 0, n 2 a i (i = 2,..., n) (iv) a = 0, n 2 a i (i = 2,..., n) [[a ], [a 2 ],..., [a n ]] (i) (iii) (positive) (ii) (iv) (negative) n (i), (ii), (iii), (iv) [[a ], [a 2 ],..., [a n ]] (canonical form) (alternating) (alternating diagram) ( ) 3-8 a 3 >0 a >0 a 2 > (two-bridge link) Bankwitz Schumann [5] a, a 2,..., a n Z t ( 3-4 ) T = [[a ], [a 2 ],..., [a n ]] a i, a i ( a i a i < 0 ) i {2,..., n} r(t ) n T = [[a ], [a 2 ],..., [a n ]] l(t ) 44

45 () n 3 a n 0, a 2,..., a n 0 a i T T = [[a ],..., [a i 2 ], [a i + a i+ ], [a i+2 ],..., [a n ]] r(t ) + l(t ) < r(t ) + l(t ). (2) T [ ] a 2,..., a n 0 S : (i) S T (ii) r(s) + l(s) < r(t ) + l(t ). (iii) S 2 [0] ( ) () l(t ) = l(t ) 2 T a,..., a i 2 a k a k < 0 k p a i+2,..., a n a k a k < 0 k q r(t ) r(t ) + a i 2 > 0, a i+2 > 0 a i > 0, a i+ > 0 a i + a i+ > 0 r(t ) = p + q = r(t ) r(t ) + a i > 0, a i+ < 0 r(t ) = p + q + p + q (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + 2 (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ > 0 r(t ) = p + q + p + q (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + 2 (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ < 0 a i + a i+ < 0 r(t ) = p + q + 2 r(t ) = p + q + 2 = r(t ) r(t ) + a i 2 > 0, a i+2 < 0 a i > 0, a i+ > 0 a i + a i+ > 0 r(t ) = p + q + = r(t ) r(t ) + 45

46 a i > 0, a i+ < 0 r(t ) = p + q p + q + (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ > 0 r(t ) = p + q + 2 p + q + (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ < 0 a i + a i+ < 0 r(t ) = p + q + r(t ) = p + q + = r(t ) r(t ) + a i 2 < 0, a i+2 > 0 a i > 0, a i+ > 0 a i + a i+ > 0 r(t ) = p + q + = r(t ) r(t ) + a i > 0, a i+ < 0 r(t ) = p + q + 2 p + q + (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ > 0 r(t ) = p + q p + q + (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q + (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ < 0 a i + a i+ < 0 r(t ) = p + q + r(t ) = p + q + = r(t ) r(t ) + a i 2 < 0, a i+2 < 0 a i > 0, a i+ > 0 a i + a i+ > 0 r(t ) = p + q + 2 = r(t ) r(t ) + a i > 0, a i+ < 0 r(t ) = p + q + p + q + 2 (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q (if a i + a i+ < 0) r(t ) r(t ) + 46

47 a i < 0, a i+ > 0 r(t ) = p + q + p + q + 2 (if a i + a i+ > 0), r(t ) = p + q (if a i + a i+ = 0), p + q (if a i + a i+ < 0) r(t ) r(t ) + a i < 0, a i+ < 0 a i + a i+ < 0 r(t ) = p + q r(t ) = p + q = r(t ) r(t ) + r(t ) r(t ) + (2) r(t ) + l(t ) = [0] T l(t ) l(t ) =, r(t ) = 0 T = [0] (2) (i),(ii),(iii) ( ) m 2 r(t ) + l(t ) < m 2 [0] T (i),(ii),(iii) S r(t ) + l(t ) = m 2 [0] T = [[a ],..., [a n ]] (a i = 0 ) i < n () T r(t ) + l(t ) < m a 0 a i + a i+ 0 T [0] T (i),(ii),(iii) a = 0 a i +a i+ 0 T [0] T (i),(ii),(iii) a 0 a i +a i+ = 0 i = 2 T [0] T (i),(ii),(iii) i 3 T = [[a ],..., [a i 2 ], [0], [a i+2 ],..., [a n ]] 2 [0] r(t ) + l(t ) < m T r(s) + l(s) < r(t ) + l(t ) S 2 [0] S (i),(ii),(iii) a = 0 a i +a i+ = 0 i = 2 a 2 = 0 a 3 = a +a 3 = 0 T 2 [0] i 3 T = [[a ], [a 2 ],..., [a i 2 ], [0], [a i+2 ],..., [a n ]] 2 [0] r(t ) + l(t ) < m T r(s) + l(s) < r(t ) + l(t ) S 2 [0] S (i),(ii),(iii) i = n T = [[a ],..., [a n ], [0]] 47

48 n = 2 T = [[a ], [0]] a = 0 T = [ ] T [ ] n 3 [[a n 2 ], [a n ], [0]] = [a n 2 ] + [a n ] + [0] = [a n 2 ] + [a n ] + [ ] = [a n 2] + [ ] = [a n 2] T T := [[a ],..., [a n 2 ]] { r(t ) + (n 2) (an 2 r(t ) + l(t a n < 0 ) ) = r(t ) + (n 2) (a n 2 a n > 0 ) r(t ) + l(t ) < r(t ) + l(t ) = m T r(s) + l(s) < r(t ) + l(t ) S 2 [0] S T S (i),(ii),(iii) 2 [0] T (2) ( 3-8 ) T r(t ) l(t ) 3-20 r(t ) + l(t ) = T l(t ) = k 2 r(t ) + l(t ) < k ( 3-4 ) T 2 0 r(t ) + l(t ) = k r(t ) = 0 T h r(t ) + l(t ) = k r(t ) < h T 2 [0] T r(t ) + l(t ) = k r(t ) = h T 2 [0] 2 [0] 3-20(2) T r(s) + l(s) < k S 2 [0] S T 2 [0] l(t ) = n T = [[a ], [a 2 ],..., [a n ]] a 2,..., a n Z {0} h+n = k h a i, a i i {2,..., n} a a i a i 4 48

49 I. a i > 0 i II. a i > 0 i III. a i < 0 i IV. a i < 0 i I T S := [[0], [], [a i ],..., [a n ]], U := [[a i + ], [a i+ ],..., [a n ]] V := [[a i+2 ],..., [a n ]] ai U ai+ V S ai 3-9(4) S = ([ ] + U) [] U rot + [] U rot + [] = [] + U rot = [] U = [[], U] ( ) U U U S S := [[], U] = [ [], [ a i ], [ a i+ ],..., [ a n ] ] T = [[a ],..., [(a i ) + ], [a i ],..., [a n ]] = [[a ],..., [a i ], [0], [], [a i ],..., [a n ]] ( 3-6(4)) = [[a ],..., [a i ], S] [[a ],..., [a i ], S ] 49

50 = [ [a ],..., [a i ], [], [ a i ], [ a i+ ],..., [ a n ] ] =: T i 3 a i 0 a i 0 a i 2 a i > 0 r(t ) = h r(t ) + l(t ) = (h ) + (n + ) = h + n = k T a i = T = [ [a ],..., [a i ], [], [0], [ a i+ ],..., [ a n ] ] ( ) a i 2 { h ( ai r(t a i+ > 0 i < n i = n ), ) = h 2 (a i a i+ < 0 i < n ) { (h ) + (n + ) = k ( ai r(t ) + l(t a i+ > 0 i < n i = n ), ) = (h 2) + (n + ) = k (a i a i+ < 0 i < n ) T a i = T = [ [a ],..., [a i 2 ], [0], [], [0], [ a i+ ],..., [ a n ] ] = [ [a ],..., [a i 2 + ], [0], [ a i+ ],..., [ a n ] ] =: T r(t ) = h 2 { h ( ai a i+ > 0 i < n i = n ), (a i a i+ < 0 i < n ) l(t ) = n { k 2 ( ai r(t ) + l(t a i+ > 0 i < n i = n ), ) = k 3 (a i a i+ < 0 i < n ) T II T S := [[], [a i ],..., [a n ]], U := [[0], [a i + ], [a i+ ],..., [a n ]] V := [[0], [a i+2 ],..., [a n ]] 3-3 S = [] + (u [ ]) [] U rot = ( ) ( ) [] U [] + U = [[0], [], U] 50

51 ai ai+ V S U ai U U U S S := [[0], [], U] = [ [0], [], [0], [ a i ], [ a i+ ],..., [ a n ] ] = [ [0], [ a i ], [ a i+ ],..., [ a n ] ] ( 3-6(4)) T = [[a ],..., [(a i ) + ], [a i ],..., [a n ]] = [[a ],..., [a i ], [0], [], [a i ],..., [a n ]] = [[a ],..., [a i ], [0], S] [[a ],..., [a i ], [0], S ] = [ [a ],..., [a i ], [0], [0], [ a i ], [ a i+ ],..., [ a n ] ] = [ [a ],..., [a i ], [ a i ], [ a i+ ],..., [ a n ] ] ( 3-6(4)) T := [ [a ],..., [a i ], [ a i ], [ a i+ ],..., [ a n ] ] 2 [0] a i 0, a i > 0 r(t ) = h r(t ) + l(t ) = (h ) + n = k T III T S := [[0], [ ], [a i ],..., [a n ]], U := [[a i ], [a i+ ],..., [a n ]] V := [[a i+2 ],..., [a n ]] 5

52 ai U ai+ V S ai 3-9(4) S = ([] + U) [ ] U rot + [ ] U rot + [ ] = [ ] + U rot = [ ] U = [[ ], U] ( ) U U U S S := [[ ], U] = [ [ ], [ a i + ], [ a i+ ],..., [ a n ] ] T = [[a ],..., [(a i + ) ], [a i ],..., [a n ]] = [[a ],..., [a i + ], [0], [ ], [a i ],..., [a n ]] ( 3-6(4)) = [[a ],..., [a i + ], S] [[a ],..., [a i + ], S ] = [ [a ],..., [a i + ], [ ], [ a i + ], [ a i+ ],..., [ a n ] ] =: T i 3 a i 0 a i + 0 a i 2 r(t ) = h r(t ) + l(t ) = (h ) + (n + ) = k T a i = T = [ [a ],..., [a i + ], [ ], [0], [ a i+ ],..., [ a n ] ] 52

53 a i < { r(t h ) = h 2 r(t ) + l(t ) = ( a i a i+ > 0 i < n i = n ), (a i a i+ < 0 i < n ) { (h ) + (n + ) = k ( a i a i+ > 0 i < n i = n ), (h 2) + (n + ) = k (a i a i+ < 0 i < n ) T a i = r(t ) + l(t ) = r(t ) = T = [ [a ],..., [a i 2 ], [0], [ ], [0], [ a i+ ],..., [ a n ] ] = [ [a ],..., [a i 2 ], [0], [ a i+ ],..., [ a n ] ] =: T { h h 2 ( a i a i+ > 0 i < n i = n ), (a i a i+ < 0 i < n ) { (h ) + (n ) = k 2 ( a i a i+ > 0 i < n i = n ), (h 2) + (n ) = k 3 (a i a i+ < 0 i < n ) T IV T S := [[ ], [a i ],..., [a n ]], U := [[0], [a i ], [a i+ ],..., [a n ]] V := [[0], [a i+2 ],..., [a n ]] ai+ V ai S U ai 3-3 S = [ ] + (U []) [ ] U rot = ( ) ( ) [ ] U [ ] + U = [[0], [ ], U] S := [[0], [ ], U] = [ [0], [ ], [0], [ a i +], [ a i+ ],..., [ a n ] ] = [ [0], [ a i ], [ a i+ ],..., [ a n ] ] 53

54 U U U S T = [[a ],..., [(a i + ) ], [a i ],..., [a n ]] = [[a ],..., [a i + ], [0], [ ], [a i ],..., [a n ]] ( 3-6(4)) = [[a ],..., [a i + ], [0], S] [[a ],..., [a i + ], [0], S ] = [ [a ],..., [a i + ], [0], [0], [ a i ], [ a i+ ],..., [ a n ] ] = [ [a ],..., [a i + ], [ a i ], [ a i+ ],..., [ a n ] ] ( 3-6(4)) T := [ [a ],..., [a i + ], [ a i ], [ a i+ ],..., [ a n ] ] r(t ) = h r(t ) + l(t ) = (h ) + n = k T 54

55 4. Conway Kauffman Lambropoulou [52] 4 - : x, y Q { }( (x, y) (0, 0)) x y Q { } (4.) x y := x + y x = x = x, 0 = 0 = 0, = 0 =, = 0 Q { } ( (x y) z = x (y z) x = y = 0 y = z = 0 x y = 0, z = 0 x = 0, y z = 0 x, y, z Q { } ) x y = x + y. (x, y Q { }, (x, y) (, )) ( ) n Z { } F ([n]), F Q { } [n] ( ) (4.2) F ([n]) := n, F := [n] n 3 [0] = [ ], = [0] [ ] F ([ ]) = = ( ) 0 = F, F ([0]) = 0 = ( = F ( (4.2) F ([n]), F F (T ) Q { } [0] [ ] ) [n] T = [[a ], [a 2 ], [a 3 ],..., [a n ]] (a,..., a n Z) F (T ) = [a, a 2, a 3,..., a n ] ( [[a ], [a 2 ],..., [a n ]] [a ], [a 2 ],..., [a n ] ) ) 55

56 4- T = [[a ],..., [a n ]] F (T ) k Z () F (T + [k]) = F (T ) + k. ( ) (2) F = T F (T ). (3) F ( T ) = F (T ). ( (4) F T ) = F (T ) [k] k. (5) i {,..., n } T := [[a i+ ],..., [a n ]] F (T ) = [a,..., a i, F (T )]. (6) i {2,..., n} a i a i = b i + c i (b i, c i Z) S = [[c i ], [a i+ ],..., [a n ]] ( ) F (T ) = [a,..., a i, b i + F (S)] = [a,..., a i, b i, 0, F (S)]. () T +[k] = [[a +k], [a 2 ],..., [a n ]] F (T ) F (T +[k]) = [a +k, a 2,..., a n ] = [a, a 2,..., a n ] + k = F (T ) + k (2) 3-6(2) T = [[0], [a ],..., [a n ]] ( ) F = F ([[0], [a ],..., [a n ]]) = [0, a,..., a n ] = T (3) 3-6(3) 2-0(3) [a,..., a n ] = F (T ) F ( T ) = F ([ a ], [ a 2 ],..., [ a n ]]) = [ a, a 2,..., a n ] = [a, a 2,..., a n ] = F (T ) (4) 3-3 T [k] = [k] + = [[0], [k], [a ],..., [a n ]] T ( F T ) = F ([[0], [k], [a ],..., [a n ]]) = [0, k, a,..., a n ] = [0, k, [a,..., a n ]] [k] = k + [a,..., a n ] = k + F (T ) = F (T ) k. (5) T F (T ) (6) F (T ) = [a,..., a i, a i,..., a n ] = [a,..., a i, b i + [c i, a i+,..., a n ]] = [a,..., a i, b i + F (S)] 2-0(4) [a,..., a i, b i, 0, F (S)] = [a,... a i, b i, 0, [c i, a i+,..., a n ]] = [a,... a i, b i, 0, c i, a i+,..., a n ] = [a,..., a i, a i,..., a n ] = F (T ) 56

57 4-2 : F (T ) F (T ) F (T ) 956 Fox [25] Fox 3 [73], [74], [84] T 2- T T T (integral coloring) T λ : { T } Z (a, b, c ) a + c = 2b b c a T Col(T ) 2- T 4 NW, NE, SW, SE λ Col(T ) NW, NE, SW, SE λ(nw), λ(ne), λ(sw), λ(se) NW NE SW SE M T : Col(T ) M(2, Z) ( ) λ(nw) λ(ne) M T (λ) = λ(sw) λ(se) M(2, Z) 2 ( ) ( ) λ(nw) λ(ne) a b = λ(sw) λ(se) c d a = b = d λ Col(T ) Col (T ) f T : Col (T ) Q { } (4.3) f T (λ) := b a b d λ Col(T ) (diagonal sum rule) λ(nw) + λ(se) = λ(ne) + λ(sw) 57

58 4-2 S, T 2- () λ Col(S + T ) S, T ( ) ( ) ( ) a b b e a e λ S, λ T M S (λ S ) =, M c d T (λ T ) = M d f S+T (λ) = c f λ T λ Col (S + T ) λ T Col (T ) λ S Col (S) f S+T (λ) = f S (λ S ) + f T (λ T ). (2) λ ( Col(T ) T T rot ( ) λ rot a b b d M T (λ) = M c d T rot(λ rot ) = λ a c λ Col (T ) λ rot Col (T rot ) f T rot(λ rot ) = f T (λ). (3) T := ( T ) vflip T (vertical reflect) ( ) λ Col(T ) ( T ) a b b a λ M T (λ) = M c d T (λ ) = d c λ λ Col (T ) λ Col (T ) f T (λ ) = f T (λ). ( ) ( ) ( ) ( ) a b b e a e () M S (λ S ) =, M c d T (λ T ) = M d f S+T (λ) = c f λ T f S (λ S ) + f T (λ T ) = b a b d + e b e f = b a e f + e b e f = e a e f = f S+T (λ). (λ T ) (2) T T T rot T ( ) T rot a b M T (λ) = M c d T rot(λ rot ) = ( ) b d λ a c d c = b a f T rot(λ rot ) = d b d c = d b b a = f T (λ) 58

59 (3) T T T ( T ) T ( ) a b b a M T (λ) = M c d T (λ ) = d c λ a c = b d λ Col (T ) f T (λ ) = a b a c = a b b d = f T (λ) T Reidemeister (RI), (RII), (RIII) 2- T φ : Col(T ) Col(T ) M T φ = M T, f T φ = f T ( ) T T Reidemeister (RI) (RI) T A A (RI) 2 A, A 2 λ : { T } Z T λ (A ) = λ (A 2 ) a a a a a λ Col(T ) λ : { T } Z { λ λ(e) (E A ), (E) = λ(a) (E = A E = A 2 ) λ Col(T ) φ : Col(T ) Col(T ) φ(λ) = λ λ M T (λ ) = M T (λ) λ Col (T ) f T (λ ) = f T (λ) M T φ = M T, f T φ = f T T T Reidemeister (RII) T (RII) 2 2 T λ Col(T ) A, B A (RII) (RII) C A, B A, B 59

60 λ : { T } Z λ(e) λ λ(a) (E) = λ(b) 2λ(A) λ(b) (E A, B, C ), (E = A ) (E = B ) (E = C ) a c a b b a b b a b b a c a λ Col(T ) φ : Col(T ) Col(T ) φ(λ) = λ λ M T (λ ) = M T (λ) λ Col (T ) f T (λ ) = f T (λ) M T φ = M T, f T φ = f T T T Reidemeister (RIII) T, T A, B, C, X, Z, W T A, B, C, Y, Z, W T C B A C B A Y X W Z W Z λ Col(T ) λ(b) + λ(w ) = 2λ(A), λ(c) + λ(x) = 2λ(A), λ(z) + λ(x) = 2λ(W ) 2λ(B) λ(c) = 2λ(A) λ(z) ) 2λ(B)+2λ(W ) = 4λ(A) λ(c)+λ(x) = 2λ(A) 2λ(B)+2λ(W ) λ(c) λ(x) = 2λ(A) λ(z) + λ(x) = 2λ(W ) 2λ(W ) λ(x) = λ(z) 2λ(B) λ(c) + λ(z) = 2λ(A) 60

61 λ : { T } Z λ(e) (E A, B, C, Z, W ), λ(a) (E = A ) λ(b) (E = B ) λ (E) = λ(c) (E = C ) λ(z) (E = Z ) λ(w ) (E = W ) 2λ(B) λ(c) (E = Y ) T c b a c b a y x w z w z φ : Col(T ) Col(T ) φ(λ) = λ λ M T (λ ) = M T (λ) λ Col (T ) f T (λ ) = f T (λ) M T φ = M T, f T φ = f T 4-4 T λ Col (T ) f T (λ) Q { } λ ( ) T T Reidemeister 4-3 φ : Col(T ) Col(T ) M T φ = M T, f T φ = f T T T ) λ Col(T ) λ := φ(λ) Col(T ) T M T (λ) = M T (λ ) λ(nw) + λ(se) = λ(ne) + λ(sw) λ (NW) + λ (SE) = λ (NE) + λ (SW) λ Col (T ) λ Col (T ) f T (λ ) = f T (λ) f T (λ) λ Col (T ) f T (λ ) λ Col (T ) T ( (( ) ) T = [a n ] + [a n 2 ] ) + [a ] [a n ] [a 2 ] 6

62 (a 2,..., a n Z {0} [a ] [0] [a n ] [ ] ) I. n = T = [a ] (a Z { }) T = [n] (n Z) a := λ(nw), c := λ(sw) T = [n] λ(ne) = (n + )a nc, λ(se) = na (n )c ) n > 0 λ a c b b2 b3 bn bn- b = 2a c, b 2 = 2b a = 3a 2c, b 3 = 2b 3 b = 4a 3c,. b n = = (n + )a nc λ(ne) = b n = (n + )a nc, λ(se) = b n = na (n )c n = 0 n < 0 m = n a c b b2 b3 bn- bn λ(ne) = b m = mc (m )a = (n + )a nc, λ(se) = b m = (m + )c ma = na + ( n + )c n < 0 λ(nw) + λ(se) = a + ( na (n )c ) = (n + )a + ( n + )c = ( (n + )a nc ) + c 62

63 = λ(ne) + λ(sw) λ Col (T ) f T (λ) = λ(ne) λ(nw) λ(ne) λ(se) = (n + )a nc a (n + )a nc ( n(a c) ) = = n na (n )c a c λ ( ) a b T = [ ] M T (λ) a, b Z M T (λ) = a b λ(nw) + λ(se) = a + b = λ(ne) + λ(sw) λ Col (T ) b a f T (λ) = f T (λ) λ λ(ne) λ(nw) λ(ne) λ(se) = b a b b = II. S T := S (n Z {0}) [n] λ Col(T ) S S λ S ( ) a b M S (λ S ) = c d a + d = b + c f S (λ ) S λ T = [n] c x x2 d λ(sw) = (n + )c nd, λ(se) = nc (n )d x3 λ(nw) + λ(se) = a + ( nc (n )d ) = (b + c d) + ( nc (n )d ) = b + ( (n + )c nd ) = λ(ne) + λ(sw) xn xn- λ Col (T ) f T (λ) = λ(ne) λ(nw) λ(ne) λ(se) = b a b ( nc (n )d ) = b a (b d) + n(d c) k := b a b d = d c Q { } λ b d f T (λ) = d c (b d) + n(d c) = n + b d d c = n + k f T (λ) λ Col(T ) 63

64 III. S T := S + [n] (n Z {0}) λ Col(T ) S S λ S ( ) a b M S (λ S ) = c d a + d = b + c f S (λ ) S λ I λ(ne) = (n + )b nd, λ(se) = nb (n )d λ(nw) + λ(se) = a + ( nb (n )d ) = (b + c d) + ( nb (n )d ) = ( (n + )b nd ) + c = λ(ne) + λ(sw) λ Col (T ) f T (λ) = λ(ne) λ(nw) λ(ne) λ(se) = (n + )b nd a (n + )b nd ( (b a) + n(b d) ) = nb (n )d b d k := b a Q { } λ b d f T (λ) = (b a) + n(b d) b d = n + b a b d = n + k f T (λ) λ Col(T ) 4-4 ( ) T Col (T ) T NW NW T t f(t) := f T (λ) Q { } (T t λ Col (T )) T λ f(t) t ( B 3 ) 64

65 ( ) T t T T Reidemeister 4-3 φ : Col(T ) Col(T ) f T φ = f T 4-4 f T (λ ) λ Col (T ) ( ) f T (λ) = (f T φ)(λ) = f T (φ(λ)) λ Col (T ) f T (λ) = f T (φ(λ)) t T 4-6 (Kauffman and Lambropoulou [52; Theorem 4] ) t f(t) (),(2),(3),(4) T f(t) = F (T ) () t k f(t + [k]) = f(t) + k. ( ) (2) t f = t f(t). (3) t f( t) = f(t). ( (4) t k f t ) = f(t) [k] [k]. ( ) () (4) 4-4 k Z { } ( ) f([k]) = k, f = T [k] k f(t) = F (T ) ) T = [[a ], [a 2 ],..., [a n ]] f(t ) = f T (λ) ( λ T Col (T ) ) n = f([a ]) = a = F ([a ]) n = 2 () 3-0 ( f([[a ], [a 2 ]]) = f [a ] + ) [a 2 ] ( ) = f [a 2 ] + [a ] ( ) = f + a = + a = [a, a 2 ] = F ([[a ], [a 2 ]]) [a 2 ] a 2 n 3 n > k [[b ],..., [b k ]] f([[b ],..., [b k ]]) = F ([[b ],..., [b k ]]) T = [[a ], [a 2 ],..., [a n ]] T = [[a 3 ], [a 4 ],..., [a n ]] ( ) f(t ) = f [a ] + [a 2 ] + T ( ( = f [a ] + [a 2 ] T )) ( 3-3) ( ) = a + f(t ) ((), (4), 3-0) a 2 65

66 ( ) = a + F (T ) a 2 ( ) = F ([[a ], [a 2 ], [a 3 ],..., [a n ]]) ( 4- ) = F (T ) T (2), (3) f(t) = F (T ) 4- F (3) T t T = ( T ) vflip λ Col (T ) T λ 4-2(3) f T (λ ) = f T (λ) = f(t) ( T ) vflip T ( 3-9) 4-5 f( t) = f T (λ ) f( t) = f T (λ ) = f(t) (2) T t T ( 3-3 ) T = ( T ) i = T rot λ Col (T ) T rot λ rot Col (T rot ) 4-2(2) f ( t ( ) (3) f = t f(t) ) = f (λ rot ) = f T rot(λ rot ) = T f T (λ) = f(t) 4-3 : Conway [0; p ] Cromwell [; Chapter 8] Bonahon Siebenmann [6; Proposition.3] Goldman Kauffman [28] Kauffman Lambropoulou [52; Theorem 3] 4-7 (Conway ) s, t F (s) = F (t) Q { } ( ) s t 4-5 F (s) = F (t) F (s) = F (t) s t { t [0], [ ] t F (t) Q {0}, F (0) = 0. F (t) = t [ ] 66

67 F (t) = 0 t [0] s, t F (s) = F (t) =: r Q {0} s t s t S, T S = [[a ], [a 2 ],..., [a m ]], T = [[b ], [b 2 ],..., [b n ]] F (s) = [a, a 2,..., a m ], F (t) = [b, b 2,..., b n ] [a, a 2,..., a m ] = r = [b, b 2,..., b n ] ( 2-2) m = n a i = b i (i =, 2,..., m) S = T s t 67

68 5. Kauffman Conway [0] t F (t) Goldman Kauffman [27,28] Kauffman Kauffman Kauffman Lambropoulou [53; 4] Goldman Kauffman [28] Kauffman 5 - : Kauffman Kauffman [48] A Laurent T Kauffman T 2 (KB) (KB2) A Laurent (KB) = A + A, (KB2) T = δ T, δ = A 2 A 2. T T Kauffman (KB) (KB) 90 (KB) = A + A T (KB) (KB2) (5.) T = d T [0] + n T [ ] (d T n T Z[A, A ]). D D = a (a Z[A, A ]) Kauffman (KB3) = D Z[A, A ] D Kauffman T (KB), (KB2) T (5.) (5.) 68

69 T R L T (state) typel R L L R typer L R 2 T n T 2 n T State(T ) S State(T ) (T S L ) (T S R ) T S := A S := (S ) S ( D 2 D 2 )[0] [ ] 2- R S T (5.2) T := T S δ S R S S State(T ) ( T (KB),(KB2) ) D State(D) S State(D) D S, S D = D S δ S S State(D) R φ : { ( ) } R Reidemeister (RII), (RIII) φ (regular isotopy invariant) 69

70 5- () D D T T (5.3) T = d T [0] + n T [ ] d T = d T (A), n T = n T (A) Z[A, A ] T (2) = A 3, = A 3. (3) A A = (A 2 A 2 ). ( ) () Reidemeister (RII) = A + A { = A A + A } + A { A + A } = A 2 + δ + + A 2 =. = Reidemeister (RIII) = A + A, = A + A = = = T T (2) () (3) (KB) (KB) 2- ( ) t S 3 2 ( ) N(t) D(t) 70

71 N(t) = t t D(t) = t = t N(t) D(t) t ( )(numerator) ( )(denominator) D(t) = N(t rot ) T () T Kauffman T Kauffman (5.4) N(T ) = d T δ + n T (5.5) D(T ) = d T + n T δ (2) d T, n T 0 ( ) () N([0]) =, N([ ]) = N(T ) = d T N([0]) + n T N([ ]) = d T δ + n T D([0]) =, D([ ]) = D(T ) = d T D([0]) + n T D([ ]) = d T + n T δ (2) D D A = D A= 5- (3) A= = D m D A= m- m {}}{ O m = A = (KB2), (KB3) A= D A= = O m A= = ( 2) m D D = 0 2- T d T = n T = 0 () N(T ) = D(T ) = 0 d T, n T 0 7

72 5-3 () 2- T, U T + U d T +U = d T d U, (2) 2- T, U T U ( ) n T +U = n T d U + d T n U + n T n U δ. d T U = d T n U + n T d U + d T d U δ, n T U = n T n U () U T U T + U = d T [0] + U + n T [ ] + U = d T ( du [0] + n U [ ] ) + n T ( du [ ] + [0] + n U [ ] + [ ] ) = d T d U [0] + (d T n U + n T d U + n T n U δ) [ ] 2 (2) () i A ω := i = exp( π 4 i) δ = A 2 A 2 = i i i + i = 0 ( ) T Kauffman T A = ω T A=ω 2- T, U (5.6) T + U A=ω = d(t )d(u) [0] A=ω + ( n(t )d(u) + d(t )n(u) ) [ ] A=ω, T U A=ω = ( d(t )n(u) + n(t )d(u) ) [0] A=ω + n(t )n(u) [ ] A=ω. d(t ), d(u), n(t ), n(u) d T, d U, n T, n U A = ω 5-2 (5.7) N(T ) A=ω = n(t ), D(T ) A=ω = d(t ) 2- T T, T in, T rot A = ω T A=ω 72

73 T T A=ω = d(t ) [0] A=ω + n(t ) [ ] A=ω () T A=ω = d(t ) [0] A=ω + n(t ) [ ] A=ω (2) T rot A=ω = n(t ) [0] A=ω + d(t ) [ ] A=ω (3) T in A=ω = n(t ) [0] A=ω + d(t ) [ ] A=ω ( ) A=ω () I. 0 T [0] [ ] T T T = T (d(t ), n(t )) (, 0) (0, ) () II. k N (k ) () T k T c L R S, U T = A S + A U T c L R U, S T = A U + A S (5.8) S = d(s) [0] + n(s) [ ], U = d(u) [0] + n(u) [ ] S = d(s) [0] + n(s) [ ], U = d(u) [0] + n(u) [ ] T = ( Ad(S) + A d(u) ) [0] + ( An(S) + A n(u) ) [ ] T = ( Ad(U) + A d(s) ) [0] + ( An(U) + A n(s) ) [ ] A = i A = A, A = A T = Ad(S) + A d(u) [0] + An(S) + A n(u) [ ] = d(t ) [0] + n(t ) [ ] k T () 73

74 (2) I. 0 T [0] [ ] T [0] T rot [ ] T [ ] T rot [0] T [0], [ ] T rot [0], [ ] (2) II. k N (k ) () T k T c L R S, U T = A S + A U T rot c L R U rot, S rot T rot = A S rot + A U rot S, U (5.8) S rot = n(s) [0] + d(s) [ ], U rot = n(u) [0] + d(u) [ ] T = ( Ad(S) + A d(u) ) [0] + ( An(S) + A n(u) ) [ ] T rot = ( An(S) + A n(u) ) [0] + ( Ad(S) + A d(u) ) [ ] = n(t ) [0] + d(t ) [ ] k T (2) (3) T in = ( T ) rot (), (2) (3) 5-2 : R = Z[A, A ] Q(R) R R (p, q) (r, s) u, v R {0} s.t. (up, uq) = (vr, vs). (p, q) p q Q(R) p q + r ps + qr = s qs (p, q, r, s R) Q(R) 74

75 2- T (5.9) Frac T (A) := n T (A) d T (A) Q(R) 5-5 Frac T (A) 2-2- T, U d T 0 d U 0 Frac T +U (A) = Frac T (A) + Frac U (A) + n T n U d T d U δ. A = ω d T 0 d U 0 ( ) Frac T +U (ω) = Frac T (ω) + Frac U (ω) d T, n T Frac T (A) Reidemeister I 5-(2) A 3 A 3 5-3() 2- T (5.0) F (T ) := i n(t ) d(t ) Q(Z[ω]) Q(Z[ω]) Q(R) F (T ) F (T ) T F (T ) = i N(T ) A=ω D(T ) A=ω n T (ω) = d T (ω) = 0 F (T ) Q { } ( ) F (T ) F (T ) (5.7) Z Z[ω] Q(Z) Q(Z[ω]) Q(Z) Q(Z[ω]) Q(Z) { 0 0 } = Q { } N(T ) A=ω = D(T ) A=ω = 0 F (T ) Q(Z) { 0 0 } 75

76 D D A=ω = up u S = S State(D) S (monocyclic) [65; Definition2] u = D S ω = i p ( ) A = ω δ = 0 D A=ω = S State(D) S = D S S, S State(D) D ( L R R L ) S State(D) S State(D) S S S D 2 ) S 2 S S S S S S D 2k p U(p) U(p) S 4 U(p) U(p) C C2 C C2 S S S U(p) 2 C, C 2 D S U(p) C C 2 D S S 76

77 C C 2 S S S C C 2 S S 2 C C 2 S S D q S p q S S S D 2(k ) S S D 2 S, S D S S D 2 2 p, q S p L q R D D S = AA (p, q A, A ) S p R q L D D S = A A (p, q A, A ) D S = D S S p R q L D D S = D S 2 S p, q L D D S = A 2 (p, q A, A ) S p, q R D D S = A 2 (p, q A, A ) D S = A 4 D S = D S 77

78 3 S p, q R D 2 D S = A 4 D S = D S, 2, 3 S, S State(D) S S D 2 S State(D) D A=ω = S State(D) S = D S = D S ( ) T 2- D S A = ω ω u, u p, q Z N(T ) A=ω = up, D(T ) A=ω = u q N(T ), D(T ) S, S S, S u, u u = ± N(T ) S, u = ± D(T ) S N(T ) S T S D 2 D 2 [ ] N(T ) S = T S D(T ) S T S D 2 D 2 [0] D(T ) S = T S 2 N(T ) A=ω 0 D(T ) A=ω 0 T S, S D 2 D 2 [ ], [0] ( 2 ) T S T S = ±i S = N(T ) S N(T ) A=ω = 0 S = D(T ) S D(T ) A=ω = 0 N(T ) A=ω 0 D(T ) A=ω 0 S = N(T ) S S = D(T ) S T S, S D 2 D 2 [ ], [0] T S D 2 D 2 [ ] S T 78

79 ( S 2 ( 2, ), 2 2 ), ( 2, 2 ), ( 2 2, ) 2 C C 2 C C 2 D 2 C C2 S D 2 [0] T S C C 2 S T S S D 2 D 2 [0] T S S T S = A ±2 T S = i ± T S = ±i T S T (5.0) F (T ) () 2- T, U d(t ) 0, n(t ) 0 F (T + U) = F (T ) + F (U). (2) F ([0]) = 0, F ([]) =, F ([ ]) = 0. (3) 2- T F ( T ) = F (T ). F ([ ]) =. (4) n(t ) = d(t ) = 0 2- T F (T ±rot ) = F (T ). F (T ±in ) = ( ) F (T ) T F = T F (T ). ( ) () 5-5 (2) (5.3) T = [0] d T =, n T = 0 F ([0]) = i n(t ) d(t ) = i0 = 0. (KB) [] = = A + A = A [ ] + A [0] 79

80 T = [] d T = A, n T = A F ([]) = i n(t ) d(t ) = ia A = i A 2 = i i = =. (5.3) T = [ ] d T = 0, n T = F ([ ]) = i n(t ) d(t ) = i 0 = 0. (3),(4) 5-4 Φ 8 = { z C z 8 = } D 5-6 (5.) D A=ω = up (p Z, u Φ 8 ) u S State(D) u = D S 5-8 ([65; Proposition 3]) T N(T ) A=ω = pu, D(T ) A=ω = qv (p, q Z, u, v Φ 8 ) ( ) p = 0, q = 0, u v = i, u v = i. N(T ) A=ω p = 0 D(T ) A=ω = 0 q = 0 N(T ) A=ω 0 D(T ) A=ω 0 p 0, q T S S N(T ) S D(T ) S = ±i 5-6 N(T ) A=ω = N(T ) S ( ), D(T ) A=ω = D(T ) S ( ) N(T ) A=ω = pu, D(T ) A=ω = qv N(T ) S = ±u, D(T ) S ± v ( N(T ) S, D(T ) S A ) N(T ) S = T S, D(T ) S = T S u v N(T ) S T S = ± D(T ) S = ± T S = ±i 80

81 6. Krebes Krebes [65; 7] 8 Kauffman 2- Krebes Q { } 2-4 Krebes 7 2- Krebes ( ) [07] ω = i, Φ 8 = { z C z 8 = } 6 - : Krebes Krebes (formal fraction) Z Z (p, q), (p, q ) Z Z (p, q) (p, q ) (p, q) = (p, q ) or (p, q) = ( p, q ). (p, q) [p, q] p q A A + 0 p q + r s T Ψ(T ) C 2 ps + qr =. qs (6.) Ψ(T ) = { (u N(T ) A=ω, ui D(T ) A=ω ) u Φ 8 } 6- ([65; Lemma 4]) T Ψ(T ) T t Ψ(t) := Ψ(T ) (T t ) t Krebes ( ) Ψ(T ) Reidemeister (RII), (RIII) Kauffman Ψ(T ) Reidemeister (RI) N(T ), D(T ) (RI) A ±3 Φ (Krebes [65; Theorem-and-Definition ]) t f(t) := Ψ(t) Z 2 A Ψ(t) Z 2 (p, q) Z 2 {(p, q), ( p, q)} ( ) 3 8

82 (i) f(t). (ii) (p, q), (p, q ) Z 2 (p, q), (p, q ) f(t) (p, q ) = (p, q) (p, q ) = ( p, q). (iii) (p, q) Z 2 (p, q) f(t) ( p, q) f(t). T t f(t) f(t ) (i) N(T ) A=ω = pu, D(T ) A=ω = qv (p, q Z, u, v Φ 8 ) 5-8 p = 0, q = 0, u = iv, u = iv p = 0 N(T ) A=ω = 0 ω := iv Φ 8 ω N(T ) A=ω = 0 ωi D(T ) A=ω = iv i qv = q f(t ) = Ψ(T ) Z 2 q = 0 ω = u Φ 8 ω N(T ) A=ω = p, ωi D(T ) A=ω = 0 f(t ) = Ψ(T ) Z 2 u = iv ω = iv Φ 8 ω N(T ) A=ω = iv p(iv) = p, ωi D(T ) A=ω = iv i qv = q ( p, q) f(t ). f(t ) u = iv ω = iv Φ 8 ω N(T ) A=ω = iv p( iv) = p, ωi D(T ) A=ω = iv i qv = q (p, q) f(t ). f(t ) (ii) (p, q), (p, q ) f(t ) u, u Φ 8 p = u N(T ) A=ω, q = ui D(T ) A=ω, p = u N(T ) A=ω, q = u i D(T ) A=ω p = 0 N(T ) A=ω = 0 p = 0 q = 0 q = 0 (p, q ) = (0, 0) = (p, q) q 0 q q = u u Q Φ 8 = {±} q = ±q (p, q ) = (0, ±q) = ±(p, q) p 0 p p = u u Q Φ 8 = {±} p = ±p p = p u = u q = q p = p u = u q = q (p, q ) = ±(p, q) 82

83 (iii) (p, q) f(t ) p = u N(T ) A=ω, q = ui D(T ) A=ω (u Φ 8 ) u Φ 8 p = ( u) N(T ) A=ω, q = ( u)i D(T ) A=ω ( p, q) f(t ) 6-3 ([65; Propositions 5, 6]) t f(t) = Ψ(t) Z 2 A s, t () f(s + t) = f(s) + f(t) (2) f(t) = [p, q] f(t rot ) = [ q, p], f( t) = [ p, q], f(t in ) = [q, p] ( ) f(t) 6- () f(s) = [p, q], f(t) = [r, s] S, T s, t u, v Φ 8 p = u N(S) A=ω, r = v N(T ) A=ω, q = ui D(S) A=ω s = vi D(T ) A=ω (5.6), (5.7) D([0]) A=ω =, N([0]) A=ω = 0, D([ ]) A=ω = 0, N([ ]) A=ω = D(S + T ) A=ω = d(s)d(t ) = ( iuq)( ivs) = qsuv, N(S + T ) A=ω = n(s)d(t ) + d(s)n(t ) = (up)( ivs) + ( iuq)(vr) = iuv(ps + rq) ps + rq = iuv N(S + T ) A=ω, qs = (iuv) i D(S + T ) A=ω iuv Φ 8 (ps+rq, qs) Ψ(S +T ) p, q, r, s Z Z Z = Z 2 f(s + T ) = Ψ(S + T ) Z 2 = [ps + rq, qs] = [p, q] + [r, s] = f(s) + f(t ) (2) f(t ) = [p, q] p = u N(T ) A=ω, q = ui D(T ) A=ω u Φ 8 N(T rot ) = D(T ), D(T rot ) = N(T ) ( 5-4) ( N(T rot ) A=ω, i D(T rot ) ( ) ) A=ω = D(T ) A=ω, i N(T ) A=ω = ( uiq, uip) = ui( q, p) 83

84 iu Φ 8 f(t rot ) = [ q, p] 5-4() N( T ) A=ω = N(T ) A=ω, D( T ) A=ω = D(T ) A=ω ( N( T ) A=ω, i D( T ) A=ω ) = ( N(T ) A=ω, i N(T ) A=ω ) f( T ) = [ p, q] = ( up, i(uiq) ) = u( p, q) T in = T rot f(t in ) = [q, p] 6-4 n Z f([n]) ( ) = [ n, ], f = [, n] [n] f([n]) = [ n, ] n = 0 N([0]) =, D([0]) = N(T ) A=ω = 0, D([0]) A=ω = Ψ([0]) = { (0, ui) u Φ 8 } f([0]) = [0, ] n = N([]) =, D([]) = A = ω N([]) = A 3 = A, D([]) = A 3 = A A N([]) =, ( A)i D([]) = A 2 i = f([]) = [, ] n > [n] = [n ] + [] f([n ]) = [ (n ), ] 6-3(3) f([n]) = f([n ]) + f([]) = [ (n ), ] + [, ] = [ n, ] n = n = A = ω N([ ]) = A 3 = A, D([ ]) = A 3 = A A 3 N([ ]) =, ( A 3 )i D([ ]) = A 2 i = f([ ]) = [, ] n < [n] = [n + ] + [ ] f([n + ]) = [ (n + ), ] 6-3(3) f([n]) = f([n + ]) + f([ ]) = [ (n + ), ] + [, ] = [ n, ] n Z f([n]) = [ n, ] 84

85 n Z [n] = [n]in 6-3(6) f ( ) [n] = f([n] in ) = [, n] T (5.0) F (T ) 6-3 f(t ) A A Q(Z) f(t ) ( ) f(t ) = [p, q] F (T ) = f(t ) ( = p ) p = un(t ), q = uid(t ) (u Φ 8 ) q 5-7 z w αz αw F (T ) = i n(t ) d(t ) = i pu iqu = p q = f(t ) (z, w, α C, α 0) 6-2 : Krebes Krebes Ψ(t) t f(t) = [p, q] (6.2) Kr(t) := gcd(p, q) t Krebes (Krebes number) gcd(0, 0) = Kr(t) 2- t ( p, q gcd(p, q) ) Kr(t) Krebes Krebes Krebes Ψ(t) 2 2-5, 6 Kr(5 ) = Kr(6 ) = Ψ(5 ) = [5, 6] [9, 0] = Ψ(6 ) Kr(t) Ψ(t) ( 5, 6 [46] 6-3 ) 85

86 Ψ(t) (Krebes ) Ψ(t) Z 2 = f(t) A f(t) Q(Z) Kr(t) F (t) (Krebes ) ( ) 6-7 (Krebes) t T 2- ( (clasp)) Ψ(t) = [0, 9], Kr(t) = 9 T = [ 3] + [3] 6-3() f(t ) = f ( ) + [ 3] f ( ) [3] 6-4 f ( ) ( ) = [, 3], f = [, 3] [ 3] [3] f(t) = [, 3] + [, 3] = [ ( 3) + 3, 3 ( 3)] = [0, 9] = [0, 9] Kr(t) = gcd(0, 9) = 9 t Kr(t) = ( A -2() ) t t L := N(t) = L S 3 2 L (= ) [2; p.67] t Krebes D (6.3) D := D A=ω (5.) D 0 86

87 6-8 L D L D D L det(l) := D L (determinant) ( ) D Reidemeister (RI), (RII), (RIII) Kauffman D (RII), (RIII) D D (RI) D D D D = A 3 D, D = A 3 D A = ω A ±3 = D = A 3 D = D, D = A 3 D = D D (RI) t T T A=ω = d(t ) [0] A=ω + n(t ) [ ] A=ω (6.4) det(n(t)) = n(t ), det(d(t)) = d(t ) ( ) (5.7) 6-8 N(T ) A=ω = n(t ), D(T ) A=ω = d(t ) det(n(t )) = N(T ) = n(t ), det(d(t )) = D(T ) = d(t ) t T t Kr(t) = gcd ( det(d(t)), det(n(t)) ) (6.5) = gcd ( d(t ), n(t ) ) 87

88 ( ) f(t) = [p, q] Krebes Kr(t) = gcd( p, q ) f(t) f(t) Ψ(T ) p = u N(T ) A=ω = un(t ), q = ui D(T ) A=ω = uid(t ) (u Φ 8 ) 6-9 p = n(t ) = det(d(t)), q = d(t ) = det(n(t)) 6-3 : Krebes 6 2- (B 3, t) (prime) 2 [66], [77;.3& Proposition.5] (i) ( ) B 3 t 2 (ii) ( ) B 3 t 2 2 S B = S (B, B t) - 3 B [46] 7 6 Yamano [46] Krebes Krebes ( ) A = ω = i 88

89 6- () n N [n] A=ω = A n [0] A=ω + na n 2 [ ] A=ω [n] = A=ω na n+2 [0] A=ω + A n [ ] A=ω (2) T T (6.6) T A=ω = 2 T A=ω. (3) 0 n n = 2A n + (2n + )A n 2. Krebes + C + (5.6) 5 Krebes 5 = [2] [3] f(5 ) = f Kr(5 ) = gcd(5, 6) = ( ) + [2] f ( ) = [, 2] + [, 3] = [5, 6] [3] 6 Krebes 6 = ( [2] [] [2] ) 3-3(3) [] 6 ( ) [2] [2] [2] [2] [2] = ( [2] [2] ) in [2] [2] ( f [2] ) [2] 6-3(2) f(6 ) = f ( [2] ) + f = f([2]) + f ( ) = [ 2, ] + [, 2] = [ 5, 2] [2] (( f [2] ) in ) = [2, 5] [2] Kr(6 ) = gcd(9, 0) = (( [2] ) in ) = [, 2] + [2, 5] = [ 9, 0] = [9, 0] [2] 89

90 6 2 Krebes 6 2 = [3] [3] f(6 2 ) = f ( ) + [3] f ( ) = [, 3] + [, 3] = [6, 9] [3] Kr(6 2 ) = gcd(6, 9) = Krebes f(6 3 ) = [0, 9], Kr(6 3 ) = Krebes T, T 2 (T = ( (5 ) rot) rot ) T T2 6 4 A=ω = A T A=ω + A T 2 A=ω T, T 2 Kauffman 6- ( A=ω ) = A + A = A( A 3 ) + A { = 2 = 2 A + A } } {A [3] + A=ω A 2 + A { 3A + A 3 } + A ( A 3 ) 2 + A 3 ( 2 ) = 6A 5A 90

91 = A + A = A { A + A } + A { A + A } = A 2 + ( A 3 ) + A + A + A 2 ( A 3 ) = A 2 A 3 + A + 2A = 7A + 6A = i ( 3i + ) Kr(6 4 ) = gcd(, 3) = 4 Krebes 4 = [2] [2] f(4 ) = f ( ) + [2] f ( ) = [, 2] + [, 2] = [4, 4] [2] Kr(4 ) = 4 5 Krebes A 4 A (4 )in 5 A=ω = ω 4 A=ω + ω (4 ) in A=ω A 4 = 4 A=ω = A + A = A( A 3 ) + A { A + A } 9

92 = 4 4i. 5-4 (4 ) in A=ω = 4i + 4 = A (8i + 8 ) f(5 ) = [8, 8] Kr(5 ) = 8 6 Krebes 6 = [2] [4] f(6 ) = f ( ) + [2] f ( ) = [, 2] + [, 4] = [6, 8] [4] Kr(6 ) = gcd(6, 8) = Krebes = A + A { } = A A + A + A ( A 3 ) = A 2 4 A=ω + ( A 3 ) + 4 = 6-4 A=ω = 4 + 4A 2 6- = 2A + 3A, = 2A

93 6 2 A=ω = (4A 2 A 3 (2A) + 2A 2 ) + (4 A 3 (3A ) + 5) = 8i + 2 f(6 2 ) = [2, 8] Kr(6 2 ) = gcd( 8i, 2 ) = Krebes = A + A ( ) = = = = A 3 = A 3 = A 3 { ( )} 6- = 3A A, = A + 3A, = 2 + A 2, = A + A = ( A + A ) ( 3 + 4A 2 ) ( (5.7)) 93

94 = 7A + 4A 3, {( ) = A 3 A + A ( )} 5A + 3A 3 = 5A + 8A ( (5.7)) 6 3 A=ω = = A + A = ( A(7A ) + A (5A) ) + ( A(4A 3 ) + A (8A ) ) = 2 2i f(6 3 ) = [2, 2], Kr(6 3 ) = gcd( 2, 2i ) = Krebes 6 2 = 4 [2] f(6 2 ) = f(4 ) + f ( ) = [4, 4] + [, 2] = [2, 8] [2] Kr(6 2 ) = gcd(2, 8) = 4 t Kr(t) det ( D(t) ) det ( N(t) ) Ψ(t) [4, 4] [5, 6] [8, 8] [9, 0] [6, 9] [6, 8] [0, 9] [2, 8] [, 3] [2, 2] [2, 8] 94

95 6 Krebes Ψ(t) 6-2 S Krebes , 6, , [46] 7 2- Krebers 5, 6, , Jones (Jones 8-3 ) 2- ( ) t W (t) := N ( t + ( t) ) t t 2- W (t) W (t ) W (t) = t - t Jones W (t) Jones t t ( ) 2- V W (t) (t) = V W (t )(t) 6-4 : 7 Krebes 7 [46] 95

96 Krebes Krebes [46, 07] 96

97 t Kr(t) det(d(t)) det(n(t)) Ψ(t) 7 7 [7, ] [3, 4] [, 2] [7, 2] [7, 2] [7, 5] [3, 5] [7, 0] [9, 4] [7, 20] [6, 9] [5, 3] [5, 20] [9, 8] [4, 2] [5, 2] [24, 2] [0, 5] [4, 6] [6, 2] [6, 2] [6, 2] [6, 6] [6, 6] [20, 20] [4, 8] [8, 8] [24, 6] Krebes 7 2- {7 2, 7 3, 7 4 } {7 5, 7 6 } 7 2, 7 3, [46; p.38] 7 2, , , 7 3, (7 2 ), (7 3 ), (7 4 ) (7 3 ) = [ ], (7 4 ) = [3] (7 2 ) (7 2 ) A=ω 97

98 = A + A = A( A 3 ) 2 + 2A = 3A (7 2 ), (7 3 ), (7 4 ) Krebes t Kr(t) det(d(t)) det(n(t)) Ψ(t) (7 2 ) [3, 0] (7 3 ) 0 [, 0] (7 4 ) 3 [, 3] 7 2, 7 3, Krebes 7 5, , 7 6, [ 3] [2] (7 5 ), (7 6 ) 7 5, (4) (3) Krebes t Kr(t) det(d(t)) det(n(t)) Ψ(t) (7 5 ) 2 [, 2] (7 6 ) 3 [3, ] Krebes 7 2- Krebes 2- Krebes 98

99 7. ( ) [5] 7 - : (rational link) 3-7 n a a2 an-2 an- an n a a3 an- a2 an n = n n (n 0), (n < 0) n T = [[a ], [a 2 ],..., [a n ]] N(T ) N(T ) C(a, a 2,..., a n ) Conway (Conway s normal form) [2], [43] [59; 2, ] 7-2 : 2 ( [35; 7 3] [5; Lemma ]) x, x 2, [x, x 2,..., x n ] ( ) ( ) F n, G n Z[x,..., x n ] 99

100 . [x ] = x = x F = x, G =. [x, x 2 ] = x + x 2 = x x 2 + x 2 F 2 = x x 2 +, G 2 = x 2. [x, x 2, x 3 ] = x [x 2, x 3 ] + [x 2, x 3 ] = x x 2 x 3 + x + x 3 x 2 x 3 + F 3 = x x 2 x 3 + x + x 3, G 3 = x 2 x 3 +. [x, x 2, x 3, x 4 ] = x [x 2, x 3, x 4 ] + [x 2, x 3, x 4 ] = x x 2 x 3 x 4 + x x 2 + x x 4 + x 3 x 4 + x 2 x 3 x 4 + x 2 + x 4 F 4 = x x 2 x 3 x 4 + x x 2 + x x 4 + x 3 x 4 +, G 4 = x 2 x 3 x 4 + x 2 + x 4. (7.) [x, x 2,..., x n ] = [x, [x 2,..., x n ]] = x [x 2,..., x n ] + [x 2,..., x n ] = x F n (x 2,..., x n ) + G n (x 2,..., x n ) F n (x 2,..., x n ) F n = x F n (x 2,..., x n ) + G n (x 2,..., x n ), G n = F n (x 2,..., x n ) G n F n F n = x F n (x 2,..., x n ) + F n 2 (x 3,..., x n ) 7- n N F n+ = x n+ F n + F n, G n+ = x n+ G n + G n ( ) n = 2 n 2 F n = x n F n + F n 2, G n = x n G n + G n 2 F n := F n (x 2,..., x n ), G n := G n (x 2,..., x n ) 00

101 (7.) k N F k+ = x F k (7.2) + G k, G k+ = F k [x, x 2,..., x n+ ] = x F n + G n F n ( (7.) ) = x (x n+ F n + F n 2 ) + x n+g n + G n 2 x n+ F n + F n 2 ( ) = x n+(x F n + G n ) + x F n 2 + G n 2 x n+ F n + F n 2 = x n+f n + F n x n+ G n + G n ( (7.2)) F n+ = x n+ F n + F n, G n+ = x n+ G n + G n F n F n (x n, x n 2,..., x ) = F n (x, x 2,..., x n ). ( ) n =, 2 n 2 n, n n + 7- F n+ = x n+ F n (x n,..., x ) + F n (x n,..., x ) ( ) (7.) F n+ (x n+, x n,..., x ) = x n+ F n (x n,..., x ) + G n (x n,..., x ) = x n+ F n (x n,..., x ) + F n (x n,..., x ) ( ) F n+ = F n+ (x n+, x n,..., x ) 7- ( ) ( ) ( ) Fn G n xn Fn G n (7.3) =. 0 F n G n F n 2 G n 2 ( ) ( ) ( ) ( ) ( ) Fn G n xn xn x3 F2 G 2 =. F n G n F G ( ) F2 G 2 = F G ( ) x x 2 + x 2 = x 0 0 ( ) ( ) x2 x 0 0