Classification of knots with unknotting number 1. Is there a classification of knots with unknotting number=1?
I can think of one infinite family at least. That is, twist the unknot $n$ times then join the two ends together.
In fact, I think maybe one could twist the unknot into a string and join the ends together to make a circle. And then form any knot with this circle and it will have unknotting number=1.
So it seems like the classification of knots with unknotting number 1 is paradoxically just as difficulat as the classification of all knots!
 A: Your first family are called twist knots, and your second family, twist knots with a knot tied into them, are called Whitehead doubles.  Every Whitehead double is a genus 1 unknotting number 1 knot, and Scharlemann and Thompson showed the converse that genus 1 unknotting number 1 knots are Whitehead doubles:
Scharlemann, Martin; Thompson, Abigail, Link genus and the Conway moves, Comment. Math. Helv. 64, No. 4, 527-535 (1989). ZBL0693.57004.
While classifying unknotting number 1 knots might be difficult, maybe some good news is that unknotting number 1 knots are prime:
Scharlemann, Martin G., Unknotting number one knots are prime, Invent. Math. 82, 37-55 (1985). ZBL0576.57004.
(Some context of this result is that there is the long-standing question of whether $$u(K_1\mathbin{\#}K_2)=u(K_1)+u(K_2).$$  So, if $u(K_i)\geq 1$ for each $i$, we at least have $u(K_1\mathbin{\#}K_2)\geq 2$.)
In general, every unknotting number 1 knot can be put in the following form.  Take an unknot, glue one end of a long strip to it, tie it up in some way, then take a small part of the other end and have the unknot pass through the strip transversely; this is called a clasp band.  Here are a few examples:

A related invariant is the clasp number, which is the number of such bands it takes to present a knot where the strips do not pass through the interior of the unknot, so clasp number is an upper bound for unknotting number.  For example, $3_1$ and $4_1$ have clasp number 1, but $6_2$ and $6_3$ have clasp number 2.  A clasp number 1 knot is the same thing as a Whitehead double.
(Something I wish I knew is whether there is a nice set of moves on these clasp bands that go between any two such presentations, especially a set that does not increase the number of clasp bands.)
