# Knots and Braids

• John Stillwell
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 72)

## Abstract

We have seen (4.2.7) that the (m, n) torus knot has group
$${G_{m,n}} = \left\langle {a,b;{a^m} = {b^n}} \right\rangle$$
. It is obvious that Gm,n = Gn,m, which reflects the less obvious fact that the (m, n) torus knot is the same as the (n, m) torus knot. Gm,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.

## Keywords

Word Problem Finite Order Covering Space Lens Space Cyclic Covering
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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