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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?

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2 Answers 2

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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

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  • $\begingroup$ @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. $\endgroup$ Commented Oct 31, 2019 at 4:42
  • $\begingroup$ Maybe, maybe not. Adams doesn't know everything. --Ken Perko, [email protected]. $\endgroup$
    – user43408
    Commented Mar 8, 2022 at 21:23
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For what it's worth, the Wikipedia page on pretzel links states that

The hyperbolic volume of the complement of the (−2,3,8) pretzel link is 4 times Catalan's constant,

which is itself described as

arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven.

The Whitehead link also has this complement volume.

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  • $\begingroup$ Two examples are slim evidence on which to base a conjecture. --Ken Perko, [email protected] $\endgroup$
    – user43408
    Commented Mar 8, 2022 at 21:26

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