3-4 point, high approach, inside contact, solid connection
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The canonical joseki for the solid connection at is as shown, with making a three-space extension from a pillar? of two stones, completing the joseki. Here the low extension is shown, may also be played at the high extension at a. The low extension emphasizes territory, while the high extension emphasizes influence. See fourth line vs. third line for more discussion on this aspect. The "White extension" section below discusses some aspects of these extensions.
Depending on the left side, may also be played at b or c, each with its pros and cons. The advantage of here is that it finishes the position in one move. See the "Black extension" section below for more.
In older days, at d is also seen (Takahashi Shigeyuki (B) vs. Go Seigen 1936-04-14), but presently it is no longer considered to be joseki. See 3-4 point, high approach, inside contact / discussion about why d is considered to be too wide.
In special cases, may also be played as a pincer at or around . See the section on "Black pincer variant" below for more.
With a checking extension at , it is possible for Black to invade this three-space extension if White does not defend at a. Usually the result of this invasion is that Black connects back with , while White has a wall. However, this wall is without base, so depending on the remainder of the board, the White group might actually become a floating group.
makes miai of a and b. Alternatives are at immediately and at a for ko.
In particular if Black has support at the circled points, this invasion can be devastating.
dnerra: Hmm, it looks to me like White has many other options, and the chosen one is far from optimal. There are more choices, but looks like a mistake unless White really really has to protect territory in the upper right.
When I have seen /, was played at c (cf. 'deep invasion' dia. below), making miai of b and d. Far more frequent than seems to be c directly however:
tderz is not as optimal as c (see  below), because a black move around e is missing.
IMO, there is nothing wrong with a little running fight , starting with .
If now black f, white g, then a partial result will be that the area around e will be influenced more by White and Black's influence around p will be reduced.
If B goes for another option, starting with sabaki tactics k-l-m or so, it's just a different game. Important is simply that one is not too obsessed by the idea The area around must always be White's area, rather stay flexible and be ready to exchange. tderz
is very effective. It threatens to link up at a. If White prevents this with , the fight can become very difficult for White after . Often, White just defends with at b. After that, Black can treat as kikashi if he likes.
The one-space jump of finishes the joseki in one move, with no intention of further expansion along the left side.
The diagonal move at implies an interest in further expanding the left side with a move at . Later, Black can play at a for further expansion.
As the stones form diagonals that are compromised by , in this variation is a common probing move, which aims at Black's cutting points at a and b. As Black can only defend only one of them, White can then decide a suitable follow-up. Note that the timing of and its follow-up moves is important for achieving the best effect.
The knight's move of also aims at further expansion near . Without , a move at White a will expose the weakness of the knight's move.
Black can play at or a if White's extension frustrates Black plans at the top. (This is bad locally, but may be playable overall.) White's compensation is to fence in Black and create substantial thickness. For both players, knowing how to handle White's thickness is key to playing this variation.
A word about : the connection can also be at b. is a little better than c. (Compare with getting ahead with a one-point jump.)
If Black plays atari at instead ...
... and capture a big corner. at a would be soft. The stone loses half its purpose.