(Just to be precise, in this question, the word "knot" means "ambient isotopy class of a (EDIT: smooth) knot in $S^3$.") A knot in $S^3$ is called prime if it is not the [connected sum](http://en.wikipedia.org/wiki/Connected_sum#Connected_sum_of_knots) of two other non-trivial knots in $S^3$. Clearly any knot is a sum of prime knots, and it is a theorem that this decomposition is unique. One downside of this is that there are infinitely many prime knots (for example, all non-trivial torus knots are prime). Here's a vague version of the question - Is there a way to trade off the uniqueness result (and add finitely many operations) in exchange for starting with a finite list of knots?


To try to make my question a little more precise, I'll define an "operation" as a function that takes a list of knots as input and outputs a finite list of knots. I'm not sure how to say this well, but I'd like to avoid very dumb operations like "fix an ordered list of all knots $K\_1,K\_2,\ldots$ and if $K\_i$ is input, output $K\_{i+1}$." However, I am interested in dumb answers, just not as dumb as that :-)



> 
> Is there finite list of knots $L$ and a finite list of operations $O$ on knots with the property that if $S$ is the smallest set of knots which contains $L$ and is closed under the operations $O$, then $S$ contains all knots?
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> 
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For example, the first paragraph, phrased in this language, is 
  
L = all prime knots
  
O = input two knots and output their connected sum
  
S = all knots



Is there a procedure for obtaining all knots in S^3?
If you allow yourself to consider knotted trivalent graphs, instead of just knots, then you can start with just the tetrahedron and Mobius bands, and use the operations unzip, bubbling, and connect sum to get all knotted trivalent graphs. This result is due to Dylan Thurston and Dror Bar-Natan and is written up by Dylan in a [GT monograph](http://arxiv.org/abs/math/0311458). Their motivation was the lack of a good candidate for an answer to your question which stayed entirely in the language of knots. However for many situations KTGs are just as good as knots (e.g. click on a random link at [Dror's wiki](http://katlas.math.toronto.edu/drorbn/index.php?title=Main_Page)).


