# Reference on representations of knot groups

Recently, I was studying knot groups and I wanted to learn some more material about them (e.g. their 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 knot groups, their subgroups and their representations (and related topics to the knot groups)?

• It would be difficult for a textbook to cover it entirely, as the subject is a little too thick for a not-too-huge textbook. Is there something in particular you would like to learn? Commented Sep 11, 2014 at 13:15
• @RyanBudney: I am interested in its representations in the symmetric group and the dihedral group. Commented Sep 11, 2014 at 14:05
• But what kind of things do you want to know about these representations? Commented Sep 11, 2014 at 20:47

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.

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.

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).

$SL_2(\mathbb C)$ representations of knot groups have been studied a lot. One classical reference with some very nice theorems is "Plane Curves Associated to Character Varieties of 3-Manifolds," by Culler, Cooper, Gillet, Long, and Shalen. (See also other papers of Culler and Shalen.) They define the $A$-polynomial, a knot invariant which Dunfield and Garafoulidis later showed detects the unknot (using a hard theorem of Kronheimer and Mrowka).

Another reference is "$SL_2(\mathbb C)$ representations of finitely presented groups," by Brumfiel and Hilden. One of their motivations for writing the book was studying representations of knot groups, so they discuss knot groups quite a bit.

I would second the excellent suggestions of Neil Hoffman above. For somewhat more recent literature, I can suggest two survey articles:

1. Peter Shalen, Representations of 3-manifold groups. This deals with character varieties.

2. Stefan Friedl and Stefano Vidussi, A survey of twisted Alexander polynomials. An application, and an active research topic.

They are for 3-manifold groups, not only knot groups, however.

It's a bit dated, but I found Neuwirth's book very useful, containing useful material not easily found in other sources:

Neuwirth, L. P. (1965). Knot groups (No. 56). Princeton University Press.

A key result on knot groups, not mentioned in other answers (and not mentioned in Burde-Zieschang), is the characterization by Johnson and Livingston of peripherally specified homomorphs of knot groups, substantially sharpening the results mentioned in Zhiyun Chen's answer:

Johnson, D., and Livingston, C. Peripherally specified homomorphs of knot groups, Transactions of the AMS, 311 (1989) 135-146.

On a personal note, somehow if feels to me that the topic of knot group representations has fallen out of fashion; but I think that there is a huge amount of interesting work to be done extending and refining these results, and I hope that the field makes a comeback.

Representations on dihedral groups and the symmetric group S4 are explained in my 1975 and 1976 papers: "Octahedral knot covers" and "On dihedral covering spaces of knots." They are easy to see, and use to construct explicitly the various covering spaces (for the various permutational representations of those abstract groups), --Ken Perko ([email protected])

• Original sources date back to Heegaard's 1898 Thesis and Wirtinger's 1905 explanations of how to construct those covering spaces of knots from homomorphisms of a knot group onto finite permutation groups. Fox looked at this in his 1956 Free Differential Calculus III paper, calculating linking numbers between branch curves for the 3-colored covers of a couple of examples. At his suggestion , I spelled this out in detail in my 1964 Princeton senior thesis, as he explains in his 1970 Canadian Journal paper. -Ken Perko Commented May 10, 2021 at 21:43