unknotting

An unknotting is a sequence of pass-throughs to unknot it, and the unknotting number is the minimal number of pass-throughs among all unknottings.

Unknotting sequences are actually recursively enumerable, but it's unknown whether there is a finite-time algorithm to compute unknotting number!

It's possible that this structure is a good way to derive that the unknotting number is 5.

Exhibiting a unknotting of K with n moves only gives you an upper bound u(K) <= n.

But seeing the shockingly complicated unknotting procedure here makes it much clearer why it wasn't!

And that freedom, at least to me, is intuitively connected to the unknotting number.

I spent the better part of the summer during grad school trying to prove additivity of unknotting numbers.

This hype is not just a hype it's the unknotting of the ml knot.

That link appears to be about unknotting any embedding of the circle into R^4.

Only then do you get a the special unknotting that falsifies the conjecture.

The general idea is represent an unknotting using a certain kind of surface, which can be more restrictive than a general unknotting, and then maybe that version of unknotting can be proved to be additive.

It proves that when you add knots together you can't just add together their unknotting numbers and expect to get a correct answer.

Maybe I'm really dumb, but it should be obvious that replacing a section of rope in one knot with another, is intuitively not going to simply "add the unknotting numbers"

There is a notion of "positive" and "negative" crossings, so you could define the positive and negative unknotting numbers by asking how many of each you have to swap.

Unknotting also involves moving the strings around to add or remove crossings, without performing crossing changes (if you take a loop and twist it into a figure eight, you've moved the strings and created a crossing but you haven't cut the strings and performed a crossing change).