I would greatly appreciate if I could get some help in clarifying my understanding. (This is a special topic I am studying as a 2nd year University student - I haven't taken topology yet - so please keep this in mind if answering the question, I probably won't understand the answer if it requires a lot of background in topology)
I'm reading The Knot Book by Colin C. Adams, and from my understanding of the introductory chapters, when two knots are 'composed' together - they can form different knots depending on where the arc is removed, but we must consider the orientation of the knots.
There are two cases:
(1) If the orientations of both knots match up - then no matter where we remove the arc to make the composition, we could slide one of the knots to any other projection of the same knot. Which seems to make intuitive sense.
(2) If the orientations of the knots clash, then composing the two knots from different arc openings can yield different knots.
(Q1) For the 2nd case, Is this because the orientations clash and you can't 'slide' the knots around?
(Q2) The book does say that if at least one of the knots are invertible - then it can always be deformed to reverse the orientation so that it matches the other knot. But when would this occur? Surely it would be before the two knots are composed. So would that mean - reversing the orientation yields a distinct knot, and not reversing the orientation potentially yields another distinct knot?
Would that also mean if we disregard orientation - that all compositions of two knots would yield a single distinct knot?
Many thanks in advance.
Extra Clarification Added:
Just to further clarify, my confusion lies in the order of the operations:
Say I have two distinct knots, I can compose it on one side, or I can say, flip one of the knots upside down and compose it from the other side.
The fundamental question is this: I want to know whether I can turn one of these projections to the other.
The book says that if one of the knots is invertible then yes one projection can be made into the other. Does that mean the knot can be made to go in the other direction after it is connected (from the other side)?
Because if it requires it to be inverted before it is connected, then isn't that not answering my question? (I'm asking if doing a particular thing (flipping a knot and putting it on the other side) will yield the same result as the first thing, and I am being told if you do something else (inverting a knot) it will give the same result - but that's not my question?)
Does that make sense? Or have I got the idea entirely wrong?