The loops I am talking about are unknots (rubber bands in 3-space).
What is the mathematical difference between a straight line that passes through the "inside" a closed loop in 3-space versus a straight line that exists on the outside of the closed loop?
Is there a simple way to characterize their differences?
Consider the following projection of an unknot as a concrete example:
How can I easily tell, without unravelling or deforming the unknot, if a line perpendicular to the screen passing through any given point in the above image will pass through the "inside" of the unknot? (i.e. what is a geometric way to know if a line will pass through the inside of the loop?)
Let's say a line is through a knot if it's not possible to disentangle the knot from the line in the following sense:
For this to not be trivial, the line must be infinitely long, otherwise you can just drag the knot to the end and off the line, then drag it back. (Equivalently, we can consider manipulating the knot diagram itself while having the knot avoid the blue line that goes perpendicularly through the diagram.)
There's a topological trick we can use to reimagine this problem. The main observation is that any disentangling will only ever make use of a some bounded region of the infinite space. This means that if it's possible to disentangle the knot, it's also possible to disentangle the knot when we connect up both "ends" of the infinite line to form a large loop (so, truncate the line far enough away from the region in which we disentangle the knot, then take some unknotted path outside the region and connect things up).
Through some topological reasoning, it turns out the question of whether a loop goes through a knot is equivalent to the original one. In particular, whether a loop that goes perpendicularly through a diagram exactly twice (once through the periphery and once anywhere else) can be disentangled.
This turns out to be a difficult problem. In the language of knot theory, you're asking whether there's a simple algorithm that can determine whether a 2-component link whose components are unknots is a nonsplit link -- the answer to this question is exactly whether or not the line is going through the knot, according to the definition we chose at the beginning.
There are some invariants you can calculate that are partial answers. The comments suggest the winding number, which is precisely the linking number of the two components of the link. If it's nonzero the line is definitely going through, but if it's zero it might or it might not. For example, the following is the Whitehead link, and its linking number is 0, but you cannot disentangle the knot:
If you allow the line to deform and cross through itself(!) while attempting to disentangle the knot, then if the linking number is 0 it is actually possible to disentangle the knot. I'm not sure if this really captures the idea of "going through" though. It does, however, capture the essence of the idea in Thurston's "Knots to Narnia", but where every time you go around the knot you pass through a portal -- linking number 0 means that if you travel along the line you will end up back in your original universe.
While I said disentangling is a difficult problem, it's definitely not impossible, and there is even an exponential algorithm to calculate whether the link can be disentangled. It uses the following property:
This disk is a geometric representation of a disentangling of an unknot. The idea is that if you had such a disk, if you give it polar coordinates $(r,\theta)$ you can consider each of the knots with $r=c$ for varying $c$. By continuously varying $c$ to be smaller and smaller, you are continuously deforming the knot, and for small enough $c$ the knot becomes arbitrarily unknotted (and definitely not encircling the blue loop in any way). Conversely, if it is possible to disentangle the unknot from the blue loop, then that disentangled unknot bounds a disk of the right type, and then by using the isotopy extension property you can drag the disk along while reversing the disentangling and get the right type of disk for the original configuration.
A little more detail about the algorithm I have in mind: there's a finite triangulation of the space outside the knots, and if one of these disks exists, there is also one that can be described in "normal coordinates" using normal surface theory. There are some integer linear programs you can set up to enumerate all possible disks, and we can say the blue line is through the knot if and only if all these integer linear programs return no solutions.
I guess Dynnikov's algorithm can also be used to solve the problem of whether links can be split (what I've been calling disentangled). It too is exponential time, but you might not like it because its steps involve deforming the knot in a very direct way.