Mathematical Theory of Knots and Braids (Mathematics Studies by Siegfried Moran

By Siegfried Moran

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Where h i z h i $lH = E 5 = E E and so on, modulo the relations $l(h) = $12(h) for a l l h E H. 27 SOME NECESSARY GROUP THEORY (b) We consider the s e t r of a l l elements of the form UNIQUENESS. where h belongs t o H and k i s any integer G1 ;i G2 2 We define an action of 0. on the r i g h t of r by means of the procedure i n p a r t (a) of t h i s More specifically one has t h a t G1 a c t s on the r i g h t of r as proof. , gi 1 =I Y k- 1 hl) if gi - , g, E G1 and = hg k k with hgl hl i n G k hl) i f gi k- 1 1 G1 a n d % E k 1 = 1 h1 i n GI # e # e with We impose a similar definition f o r the action of G 2 on Clearly the unit element of G1 and of G2 leave a l l the elements of r f o r a l l g1 in G1.

R2 Then L2k0 - R3ill. X 0 Product of loops R2Ro PROOF. Suppose t h a t to and G(s,t) respectively. t2 - k3 i s given by the homotopy F ( s , t ) Then by means of the homotopy H(s,t) = F(s,2t) for 0 2 t 2 4 4 5 t 2 1 G ( s , 2 t - l ) for and a l l s E I. F. 3. Suppose t h a t e is t he t r i v i a 2 loop in a topological space RESULT. X a t xoy t h a t is, e(x) eR - xo f o r a l l x i n X. = Then il f o r a l l loops R in X a t xo. Also This follows as a consequence of the following CONTINUOUS MANGE OF PARAMETER LEMMA.

A o l a -1 i s a consequence of (ul, -1 -1 -1 a u 1 a u l a) = e which i s a consequence of the relations a = u1 . a ola-l . a 2ola -2 . an-2 ala-(n-2) (premultiply by (ala ala i -' (al, a ola = -1 -1 ) e for 2 ) and 2 i 5 n 2 together with the f a c t t h a t an belongs t o the centre of Bn. The l a t t e r f a c t , as noted above, is a consequence of CHAPTER 1 20 an = (a alln-'. , an- 1 and defining relations and f o r li (ai,aj) = e - jl z 2. These relations are consequences of the following relations which hold i n Bn/B; : Ul = u2 = 6 .

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