One good book I've found on the subject is https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From.
See Thurston's Knots to Narnia and Not Knot. PolyCut is a little applet that can visualize these knotty portals.
Maths is a truly astonishing discipline begging the question, are we all just made of maths?
Zeeman was very quotable: "Technical skill is mastery of complexity while creativity is mastery of simplicity."
 Wikipedia entry - https://en.wikipedia.org/wiki/Christopher_Zeeman
 in memory of Zeeman - http://www2.warwick.ac.uk/knowledge/science/zeeman
On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.
"Spontaneous knotting of an agitated string" 
From the abstract: "We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots."
But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different simplical complexes or different CW complexes), the dinvariant will assign different values. On the other hand, if the knots are equivalent, the assigned values might not be equal.
What I want to know is: Why doesn't there seem to exist a theory of dinvariants for knots (or topological spaces)?
How would you go about showing that something is a dinvariant? You would need to take two objects that are distinct and show that their dinvariants are distinct. I.e. use the fact that a homotopy between the two knots does not exist. This is much more difficult, though it does get done. In particular, it's done whenever we have a complete classification of all objects of a certain type (say orientable surfaces, classified by the single invariant (and also dinvariant) their genus). Though generally here we would be actually approaching this from an invariant perspective but just showing that once you have enough invariants, they collectively become a dinvariant.
Dually, you would want the following: if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.
(full disclosure: lawyer who did his undergraduate degree in math, not a topologist)
> if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.
That doesn't make them useless. Dinvariants just serve a different purpose:
- invariants serve the purpose of distinguishing knots that are different
- dinvariants serve the purpose of detecting that knots that look very different are actually the same
But why don't we build a systematic theory of dinvariants (similar to the theory of invariants) to get a much better understanding of them?
Not every dinvariant needs to be an invariant. :-)
If you say two knots are equivalent, in mathematics, that means they are equal, one and the same. You can't have a tool that assigns different structures to the same thing. But I see where you are coming from. In mathematics we have the concept of a presentation (of a group or a vector space, for example). You can imagine a presentation of a knot like in a two dimensional representation (typical one) and assign an invariant to that. That wouldn't be an invariant of a knot, that would be an invariant of a knot, as you mention, but it would be an invariant of a knot plus some extra structure. Those objects are also studied in many instances in which the original is two difficult. As simple example imagine a knot plus an orientation. Invariants of that structure might be dinvariants of the knot.
As far as I know a knot is a continuous embedding of S^1 into R^3. Equal means exactly same subset of R^3. Equivalence of knots means that there is a suitable ambient isotopy between the nots (I just looked this up in wikipedia: https://en.wikipedia.org/wiki/Knot_theory#Knot_equivalence). So equality is a much stronger criterion than equivalence.
So 'equal => equivalence', but the other direction 'equivalence => equal' trivially does not hold (just apply some transition in R^3; these knots are equivalent but not equal).
It's been a while since I've studied this, but I believe it's the case that the only time you get nontrivial knots is when n = m + 2. The most well known case, of course, is when m = 1 and n = 3. But for every value of m >= 1, there are nontrivial knots in dimension m + 2. I believe it is indeed a rich area of research.
Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane.
On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told.
It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]
The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.