Josekis

The word "joseki" is borrowed from the game of Go because of the strong analogy between Hex and Go corner play. There are certain sequences of exact same moves played nearby corners of the board. Very often going into the sequence is the best play both players can do in a given position. Hence the corner moves are considered the strongest ones (by mini-max rule mentioned here) the more frequent josekis are. Knowing josekis narrows down the number of options to choose from. Nevertheless, that doesn't make player's life very easy - picking a correct joseki is as difficult as picking a correct global move. Therefore, this subject is addressed mostly to the advanced players.

Joseki's logic
When player A makes a move in the corner he dominates the territory around this corner. Assuming that he picked his best strategy, he should stick to it and always defend his domination in the area. However, that doesn't mean that player B should give up that territory. Playing somewhere else will often lead to losing all initiative around previously mentioned corner, but there is a direct counterplay for player B. A game progresses in that manner: player B counterplays, player A holds on to his domination, player B counterplays again etc... This usually lasts 2-5 moves by each player depending on how far from an edge was the first move by player A. When the final position is reached player A has a fixed escape on one side of the corner and player B has a fixed escape on the other side (second player's escape is always of shorter range).

4th line josekis
A1) Red wants an escape on bottom and Blue wants an escape on top: 

A2) Red wants an escape on bottom, Blue wants one on top and Blue knows that Red wants an escape on bottom. The move #3 is an improvement, because in some cases Blue might get an additional free escape on bottom. It is risky because if Red made an error on move #2 and in truth he wants an escape on top he can correct the error by playing a different move #4. 

B) Red wants one on top and Blue wants an escape on bottom: 

C) If both players want an escape on top this position might be reached. Blue keeps a future option of playing in the cell marked with an asterisk. This is a very risky plan by Blue, because Red has additionally got an escape on bottom. Blue should only play this joseki if he is certain that Red cannot use the escape on bottom. 

D) If both players want an escape on bottom we get an unusual position where Red plays the last move of the joseki. Blue's plan is to get a cell marked with an asterisk for free in future, so he shouldn't play there immediately. What he should do depends on the play in rest of the board. 

Other examples of josekis will be presented soon!