328

M Heusener

1 Introduction

Given two groups G and Я, а representation of G into Я is a homomorphism g G-^ H Two representations g and g' are equivalent iff they differ by an inner automorphism of Я Given a knot group G, we call a representation g G-* H abehan if its image is abelian - then it must be cyclic

2 - bridge knots and links - sometimes called 4-plats (Viergeflechte) -- were first investigated by Bankwitz and Schumann [2] where they are shown to be ing and invertible A 2-bridge knot or link ï с S^ may be described by two numbers a, ßeZ where a>0, —a<jS<a, gcd(a, ß) = I and ß = 1 mod 2 We denote the two bridge knot or hnk f determined by the numbers (a, ß) by b(a, ß) b(a, ß) is a knot (link) iff a is odd (even) For definitions, details and more information see [4, Chap 12]

2 - bridge knots are classified by their twofold branched covering - a method due to Seifert It IS easy to prove (see [4, Chap 12]) that the twofold branched covering space of b(a, ß) a S^ is the lens spaces L(a, ß) Therefore, we obtain a classification of unonented 2-bridge knots and links by the classification of lens spaces However, the general result is due to Schubert

1 . 1 Theorem (H Schubert) (a) b(a, ß) and Ъ{а, ß') are equivalent as oriented knots (or links), if and only if

a = a' and ß^=ß'mod2oc

( b ) b(a, ß) and b(a', ß') are equivalent as unonented knots {or links), if and only if

a = a' and ß^=ß'moaoi

For the proof of (a) we refer to [17] The weaker part (b) follows from the classification of lens spaces

Given b(a, ß) с S^ we denote the fundamental group of its complement by G(a, ß) Using the normal form of the 2-bridge knots or links b(a, ß) we can get a Wirtinger presentation

G ( a , ß) = <5, T\ LsS = TLsX ^s = S'T' S'' T' ' if a is odd

and

G ( aJ ) = <S, T\LsT= TLs}, Ls = S'' Г' S'' T' ' if a is even

where ei = (— 1) "" (for every real x let [x] be the greatest integer n such that n ^ x)

We denote by S^ the unit quaternions

( P , (p) = cos(i ф) + sin(i (p) P

where S^ is the 2-sphere of pure unit quaternions defined by P^ = — 1 There is a twofold covering ô S^ -^ S03(IR) = RP^, (P, (p)\-^ô{P, cp) which is a group morphism with Kernel (5) = { ± 1} (5(P, ф) is a rotation of angle cp with axis P

12 Remark It is usual to identify the unit quaternions with the group

SU2 ( ( C ) of special unitary matrices, SU2(C) = < I у

l\ - b aj

isomorphism W S -^ SU2((C) is given by W ao + a^'i + a2*j +

aä + bb=l} The