Low invasion of fourth line three space extension

 

White invades on the 3rd line, instead of splitting at a. The idea is to split with a one space jump to b or to crawl on the 3rd line for eyespace.

Link up  

Black's basic answer is to play and link up the black stones, inviting white to make life along the edge. What happens in reality depends a lot on surrounding conditions. The stronger Black's stones, the less valuable it is to connect them, in which case he will more likely allow White to jump out or even try to kill it outright.

%
%

Here's a kind of middle game joseki, arising from a san ren sei

Invasion  

When invades, there is room to both sides, so Black rather connects. If Black can block thanks to the presence of the marked stone.

Double atari  

The double atari of is a tesuji. White must live in gote with a to c. Black has sente to play in the upper left.

Although White's invasion has succeeded, it's with small life and the upper half of Black's original sphere of influence is intact, even reinforced. The skirmish has also slightly damaged White's potential at the bottom. Black comes out with sente, so this is an acceptable result.