Swan (Practical English Usage) says that
"after the commonest verbs of saying and thinking, that can be left out, particularly in a conversational style.
...
Jane thought it was time to buy some new clothes.
...
That is not so often left out in a more formal style, and there are many verbs (e,g, reply, telegraph) after which that is necessary.
He replied that we ought to invest half the profits."
I'd say that know is a member of the "thinking verbs".