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Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely covered. Also, Rolfsen talks about the fundamental group and Wirtinger presentation but not about the representations in the symmetric group or the dihedral group. So, what is a good reference for the knot group, its subgroups and its representations (and related topics to the knot groups)? |
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The references you mention are a few decades out of date, and the most studied knot group representations are those to $SL(2, \mathbb{C}).$ I don't know that there is a canonical reference for that, other than, of course, Thurston, William P., and Silvio Levy, eds. Three-dimensional geometry and topology. Vol. 1. Princeton university press, 1997. (which is a textbook, not a reference). |
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F. Gonzalez-Acuna. Homomorphs of knot groups. Ann. of Math. (2) 102 (1975), 373-377 In this paper the author studied the homomorphic images of knot groups. It was proved that a finite group is the homomorphic image of some knot group iff it is generated by the conjugates of one element. A simple proof can be found in D. Johnson. Homomorphs of knot groups. PAMS, 78 (1980), 135-138 As Ryan Budney mentioned, it is difficult for a paper or a textbook to cover this topic entirely. For some concrete groups, such as the dihedral group, there are many good references. For example the references on Wiki http://en.wikipedia.org/wiki/Fox_n-coloring would be helpful. |
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Let me add another voice to advocate that this question might be a little too broad to be answered by a specific reference. It's more of an active research area than something that fits nicely into one article or book. However, in addition to the wonderful suggestions of Zhiyun Cheng and Igor Rivin, I would suggest reading Riley's papers, especially: NONABELIAN REPRESENTATIONS OF 2-BRIDGE KNOT GROUPS By ROBERT RILEY Quart. J. Math. Oxford (2), 35 (1984), 191-208 PARABOLIC REPRESENTATIONS OF KNOT GROUPS, I, By ROBERT RILEY Proc. London Math. Soc. (3) 24 (1972) 217-242 PARABOLIC REPRESENTATIONS OF KNOT GROUPS, II, By ROBERT RILEY Proc. London Math. Soc. (3) 31 (1975) 495-512 The later two Riley articles include a good discussion of $PSL(2,F)$ representations of knot groups where $F$ is a finite field. I will concede that these are dated references, but they contain a variety of good techniques, which are useful for studying these questions. |
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