Participles are verb forms that are used as adjectives or to tell more
about the sentence as a whole.

Present Participles are as follows:

running
going
working
etc.

Perfect Participles:

having lived
having eaten
having seen etc

The negative is formed by putting 'NOT' in front:

Not knowing...
Not having seen...

Examples:

Ben WAS SITTING on the train. He had a fantastic idea.
Or:
SITTING on the train, Ben had a fantastic idea.

I DIDN'T KNOW Jane was out. I stopped by to see her.
Or:
NOT KNOWING Jane was out, I stopped by to see her.

Ed HAD READ the newspaper. He knew about the merger.
Or:
HAVING READ the newspaper, Ed knew about the merger.

We couldn't recommend Peter’s plans. We HADN'T SPOKEN to him.
Or:
NOT HAVING SPOKEN to Peter, we couldn't recommend his plans.

The subject of the sentence and the subject of the participle are generally the same, and the participle goes next to the noun it modifies.

Avoid sentences like these:

Walking down the street, a car almost hit me. ( I was walking, not the car!)

Having arrived late, there was no food left. ( The food didn't arrive late!)