This week, we're going to be discussing the attachment to the 4-4 in response to the low-knights move.

This joseki looks quite similar to the one played out last week, and with the introduction of AI, this sequence has dramatically increased in popularity and is played out more nowadays than the joseki shown last week. The purpose of this joseki is to limit black's territory in the corner and help white settle comfortably on the side.
So, for the first variant of this joseki, black plays at 7, as shown below and looks to cut white off from the outside, whilst giving up on the inside. This joseki is playable for both white and black as black trades the corner for influence, while white gets the corner and still has aji later on with the 2 stones.

The next variant that we are going to look at is when black decides to make a tigers mouth at 9, instead of descending. This variation gives black influence towards the left side, while giving up a lot of the potential corner territory, and allows white to make solid territory on the right side. Black can follow up with either A or B, and the result is playable for both black and white in this situation.

Both of these variations are playable in games IF black has space to extend his influence, and is not forced to make a 1 or 2 space extension. These variations should only be used when black has the opportunity to use the influence he had just built, and not be limited by white.
In the case that black cannot use his influence, he should play the variation given down below. Both black and white settle well. but I personally feel white has a slight advantage in this variation.

So you might have also been wondering what might happen if black decides to double-hane in this joseki. While it's uncommon, black can double-hane in this situation to preserve the corner.
The double-hane is often met by the connection of white at 8, and so black gets to keep the corner, while white settles on the right side.

In a more interesting variation, white decides to play at 8 here instead of connecting before. When black ataris at 9, white gives up the stone for influence and plays the ko at 4. White A and tenuki may be a possibility, but in my opinion,if white decides on this variation, white has to be ready to play the ko at 4. So white could follow up with A, and black must connect 3 up with the group, allowing white to take 13.

Of course, if white decides on the tiger's mouth at 8, he can always choose to connect up and avoid playing a ko.

I hope this was helpful to some! Please correct me if you see any mistakes!
Thank you