Knots and Braids

Abstract

We have seen (4.2.7) that the (m, n) torus knot has group

$$G_{m{\rm{,}}n} = \langle a,b;a^m = b^n \rangle$$

It is obvious that G _{m, n} = G _{n, m} , which reflects the less obvious fact that the (m, n) torus knot is the same as the (n, m) torus knot. G _{m, n} does not reflect the orientation of the knot in R³, since the knot and its mirror image have homeomorphic complements and hence the same group. Since Listing 1847, at least, it has been presumed that there is no ambient isotopy in R³ between the two trefoil knots (Figure 219) and the same applies to the general (m, n) knot.