One can’t test things which one doesn’t know (to) exist

I wrote a sentence:

One can’t test things which one doesn’t know to exist.

My English teacher says that is should be without to: ...know exist.

I know this phrase is part of 'concise English' but I can't explain its grammar. Can you help me? Thanks.

(a) One can’t test things which one doesn’t know exist.

(b) One can’t test things which one doesn’t know to exist.

Both are correct, but they're derived from different structures.

(a) comes from a relative clause that starts out meaning something like
- things [(such that) one doesn't know (that) these (things) exist],
and then converts into
- things [which one doesn't know exist].

(b) comes from a relative clause that starts out meaning something like
- things [(such that) one doesn't know these (things) to exist],
and then converts into
- things [which one doesn't know to exist].

Using 'thing' instead of 'things' makes the difference even more obvious:

(a) 'One cannot test a thing which one does not know exists.'

versus

(b) 'One cannot test a thing which one does not know to exist.'

So it is not just the disappearance of a 'to' but a change from conjugated verb form to infinitive verb form! Like 'something which you know is the case' versus 'something which you know to be the case'

This is possible because know can take either a that clause complement, as in (a), or an infinitive complement, as in (b). The fact that they mean the same thing and are both grammatical is normal; there are always different ways to say anything, and many of them resemble one another because they come from similar -- but not identical -- constructions.

Executive Summary: Given two sentences that mean more or less the same thing,
and have very similar structures, the way to bet is that both are equally correct.