TITLE: What is the metamathematical interpretation of knot diagrams? 
QUESTION [12 upvotes]: I am not a geometric topologist, but from looking over papers in the field, it's clear that knot diagrams are a major tool and we know how to use them in a way that is rigorous and trustworthy.  My background is in model theory and I am having trouble fitting them into that framework.  I'm hoping for pointers to references or a quick sketch of the logical status of these things.
Specific points that I'm hung up on:  knot diagrams have nice properties analogous to terms in formulas, like substitutability a la planar algebras.  On the other hand, they are relation-like, relating segments of a link to each other.  On the third hand, the Reidemeister moves seem like a set of formulas in a model-theoretic interpretation (of a "theory" of knot isotopy into a "theory" of graphs.)  Finally, there's the standard trick of calculating invariants by recursively applying certain skein relations to get to the unknot.  
In other words, I can't see that a knot diagram is always being used as a relation, term, formula, or substructure - it seems like none of these is adequate to fully describe their use.  
I can see this playing out in a number of ways:

Knot diagrams are a tool that can be completely subsumed by algebra in an algorithmic way, and are just a convenience;
There is a theorem that says proofs using diagrams can be "un-diagrammed", but it's an existence proof 
People keep finding new ways to use knot diagrams in proofs, often by decorating the diagrams with new features like orientations - reproving via alternate techniques is then a useful contribution, but always tends to work out;  e.g., we don't completely understand the metamathematics of knot diagrams, but the general shape of things is clear;
There are important theorems with no known proof except via diagrams, and nobody knows why;
It will somehow all become obvious if I take the right course on planar algebras, or o-minimal structures, or category theory;
It's subtle, but was all cleared up by Haken in the 70's;
Dude, it's just Reidemeister's Theorem, and you need to go away and think about it some more.

Community wiki, in case the right answer is a matter of opinion.
Update - Just to be clear, this is not in any way a brief for eliminating knot diagrams - quite the opposite.  Knot diagrams are honest mathematical objects, while also serving as syntax for other objects.  That seems like a ripe area for mathematical logic.
Also, I'm including partial diagrams, as in skein relations, when I use the phrase "knot diagram".

REPLY [4 votes]: I see this as a general problem in higher dimensional algebra, that there will need to be "higher dimensional rewriting".  John Baez has illustrated the higher dimensional thinking by displaying the picture
$$ ||| \;\; ||| ||  $$
$$||| \;\; ||||| $$ 
which is easily seen to illustrate $2 \times (3+5)= 2 \times 3 + 2 \times 5$ but  the 1-dimensional formula involves various conventions, and is less transparent. We have found diagrammatic rewriting useful in dealing with rotations in double groupoids (with connections), and there is a 3-dimensional rewriting argument in Section 5 of 
F.-A. Al-Agl,  R. Brown, R. Steiner,  `Multiple categories: the
equivalence between a globular and cubical approach', Advances in
Mathematics, 170 (2002) 71-118.
which proves a key braid relation (Theorem 5.2).  
So there is the interesting question of how to cope say with a 5-dimensional rewrite? Maybe computers could handle it? 
These situations could well occur in algebras with partial operations whose domains are defined by geometric conditions, and with strict axioms.