## A Survey of Knot Theory by Akio Kawauchi

(8 3 , L) such that q is close to a homeomorphism in several senses as an application of the tangle and mutation theories (cf.

1, where p+ = (0,0,0,1), p_ = (0,0,0, -1), and 52 = {(x, y, z, w) E 53 I W = O}. By putting B! = {(x,y,z,w) E 53 I W 2: O} and B~ = {(x,y,z,w) E 53 I W SO}, a 2-bridge knot or link K in 53 (cf. 3) can be presented as follows: K n {p+, p_} = ¢; and each of K n B! and K n B~ consists of two arcs which are mapped injectively into 52 by p. The arc components wi(i = 1,2) of K n B! and vi(i = 1,2) of KnB~ are the overbridges and the underbridges of K, respectively. We assume that K meets 52 in four points A, B, C, and D, where the initial point and the terminal point of Wi are A and B, respectively, the initial point and the terminal point of W2 are C and D, respectively, the initial point of Vi is B, and the initial point of V2 is D.