# Are any knot volumes known to be (ir)rational? If not, then why is the question difficult?

I'm reading C.C. Adams' The Knot Book (1994), and I'm intrigued by this assertion about the hyperbolic volume of hyperbolic knots:

Unsolved Question 2

Is any one of the volumes a rational number $$a/b$$, where $$a$$ and $$b$$ are integers? Is any one of the volumes an irrational number (not of the form $$a/b$$ where $$a$$ and $$b$$ are integers)? Amazingly enough, even though we can calculate the volume of a knot out to as many decimal places as we want, we cannot tell whether any one of the volumes is either rational or irrational.

Some of the assertions in this edition feel a bit dated, so I wanted to ask whether this assertion is still current. Is there still no knot whose complement's volume has been determined to be either rational or irrational? If there is, then which knot is it, and is it in $$\mathbb Q$$ or not? If we still don't know, are there clear reasons for why the question is hard?

• It is hard to find exact values in general, and unknown whether there are hyperbolic manifolds with rational volume at all. See math.stackexchange.com/questions/1548208/… Commented Feb 20, 2018 at 19:12

I asked Adams at a talk earlier this year if it's still true that we don't know the (exact) hyperbolic volume of a single knot. His answer was *yes." --Ken Perko

• @Leucippus I'm no knot theorist but it looks like an answer to the first question("is it still current") to me, relayed from the author of the book. Commented Oct 31, 2019 at 4:42
• Maybe, maybe not. Adams doesn't know everything. --Ken Perko, [email protected]. Commented Mar 8, 2022 at 21:23