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I'm writing a Master's thesis on knot invariants and I'm trying to chase down the original source that introduced the unknotting number and perhaps proved that it is a knot invariant. The texts I've consulted that reference the unknotting number already discuss it in terms of being an established invariant. But I can't find what source has first introduced it as a concept and proved that it is a knot invariant. I think it's obvious that Reidemeister moves don't affect the unknotting number (they preserve the number of crossings), but what is it that makes this number well-defined?

I'm a bit wary of ChatGPT, but it's pointed me in the direction of (Gordon & Luecke, 1989). I scanned through the paper but must admit found it too dense to read. Equally, the unknotting number was already an established concept by 1989 (it pops up in (Cochran & Lickorish, 1986)), so I'm curious if there isn't a different, earlier proof.

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    $\begingroup$ As I understand the definition, the fact that it is a knot invariant is trivially true. It is the smallest n such the given knot has a diagram such that the knot can be unknotted by changing n crossings. $\endgroup$
    – Tom Goodwillie
    Commented Jun 26, 2023 at 20:13

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I agree with Tom Goodwillie that it is immediate from the definitions that the unknotting number is well-defined. I think that what might be tripping you up is being taught that “knot invariants” are really diagram invariants that are invariant under the Reidemeister moves. That makes invariants like this one that cannot be computed from a given diagram in a straightforward way confusing.

Anyway, on to its origins. The earliest reference I know about is

Wendt, H. Die gordische Auflösung von Knoten. Math. Z. 42, 680–696 (1937)

It's been a very long time since I read this paper (and my German is mediocre, so it would take some effort for me to re-read it today), so I can't remember if it references anything else.

As evidence that this is the original source, Reidemeister's classic book on the subject references Wendt's paper when it introduces the unknotting number. See the beginning of Chapter 2.

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    $\begingroup$ Here's some evidence that the phrase was not coined until the late 1930s. books.google.com/ngrams/… $\endgroup$
    – Tom Goodwillie
    Commented Jun 26, 2023 at 20:52
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    $\begingroup$ @TomGoodwillie: Wow, I've never seen that google books feature before! Super cool, and pretty convincing evidence that Wendt was at least among the first to define it. $\endgroup$
    – Andy Putman
    Commented Jun 26, 2023 at 21:10
  • $\begingroup$ Amazing, thank you, I've found both the paper and a number of other resources that cite it that have been instructive to consult. $\endgroup$
    – Nobilis
    Commented Jun 27, 2023 at 11:13

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