Abstract
We have seen (4.2.7) that the (m, n) torus knot has group
. 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.
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© 1993 Springer-Verlag New York Inc.
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Stillwell, J. (1993). Knots and Braids. In: Classical Topology and Combinatorial Group Theory. Graduate Texts in Mathematics, vol 72. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4372-4_8
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DOI: https://doi.org/10.1007/978-1-4612-4372-4_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97970-0
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