Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)
Abstract
Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2007
- DOI:
- 10.48550/arXiv.0706.1807
- arXiv:
- arXiv:0706.1807
- Bibcode:
- 2007arXiv0706.1807T
- Keywords:
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- Mathematics - Geometric Topology;
- 57M27 (Primary);
- 20F38;
- 20G40 (Secondary)
- E-Print:
- 25 pages, 5 figures, to appear in JKTR. v3: example of the target groups added