Does anyone know of a good knot theory textbook for self-study? I’ve read most of The Knot Book, but it has a note saying that it won’t discuss “the fundamental group.” Is there an accessible textbook which does? Will I need to know group theory first?
-Ben
Perhaps the Boy Scout manual?
But seriously, you can have a group of knots? Trippy. What is the action?
“The fundamental group” doesn’t refer to a group of knots; rather, for each individual knot there’s an associated group. There is an action on knots (the connected sum) which takes two knots and splices them together, but it’s not a group action (technically, it’s a “monoid action”).
Regarding the OP: “Knots and Links” by Dale Rolfsen definitely discusses the fundamental group of a knot, but it (a) may not be completely suitable for self-study, and (b) is hard to find. You may have better luck finding a textbook on algebraic topology; the fundamental group of a knot is the same as the fundamental group of the topological space S[sup]3[/sup]-K (where S[sup]3[/sup] denotes the three-sphere and K is a small neighbourhood of the knot).
I’m guessing that “The Knot Book” which you’ve read is the one by Colin Adams; if that’s the case the only other book I can think of which might be suitable is “Introduction to Knot Theory” by Richard Cromwell and Ralph Fox. But again, I don’t know if it’s suitable for self-study; you may want to try and find it in a university library before you put down money for it. And I have no idea if it discusses the fundamental group.
Thanks! I’ll check it out.
-Ben
The Ashley Book of Knots is , arguably, the finest knot bible in the world.
It weighs about 3 or 4 lbs., and is so full of the finest hand drawings you wont be able to put it down.
Ashley actually cut his teeth on a whaling ship, I think, and covers ALL knots including the strictly decorative ones, along with the history of knots etc.
If you don’t fall asleep with this book on your chest , I’ll buy it off you . A truley magnificent book that you must have.