grammar - It is important that each player not know

I have just encountered the following phrase:

It is important that each player not know what the other player will do.

Why is it written here "not know" and not "not knows"? Also would it be correct to say "...player does not know..."?

Answer

This is the Subjunctive Mood. People will avoid using it by rephrasing their sentences all the time without noticing it, many times with an infinitive. In here, one would usually say it like this

It is important for each player not to know what the other player will do.

This site has good information about the subjunctive. On the site there is a list of verbs and phrases that are followed by the subjunctive (I pasted them here in case the link breaks someday).

Verbs followed by the subjunctive: to advise (that) to ask (that) to command (that) to demand (that) to desire (that) to insist (that) to propose (that) to recommend (that) to request (that) to suggest (that) to urge (that)

Source, Language Dynamics

Phrases followed by the subjunctive: It is best (that) It is crucial (that) It is desirable (that) It is essential (that) It is imperative (that) It is important (that) It is recommended (that) It is urgent (that) It is vital (that) It is a good idea (that) It is a bad idea (that)

Source, Language Dynamics

In the subjunctive, the he, she, it form of the verb takes the same verb form as the others, as you noticed in the sentence you saw.

Just keep in mind, like FumbleFingers said in his/her comment above, the subjunctive is not used very often, but it's pretty cool!