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.KeywordsWord ProblemFinite OrderCovering SpaceLens SpaceCyclic CoverThese 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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
