Construct a knot/link by fusing two n-tangles together. (A tangle matrix shows how this might look for tangles with 6 legs. But lets use 4 legs for a start as this is far simpler.)
Now, any rational link with 2n crossings can be constructed by fusing two 4-legged tangles with n crossings (is this obvious?). This is the lowest complexity in this scheme.
The next more complicated thing would be algebraic links. The Borromei rings have 6 crossings, but 4+4 is the smallest possible fusion construction.
The knot 8_18 is AFAIK the smallest "transcendent" example. I wasn't able to do better that the obvious 7+1. (Can you find any m+n-fusion such that max(m,n)<7? I think it should exist. There are about 40 tangles with 6 legs, so I didn't try brute force yet :-)
The final stage would be the Borromei double or other counterexamples to Montesino-Nakanishi. I have reasons to believe that cutting out a crossing is the best you can do (i.e. 23+1 for the Borromei double).

So I define the complexity to be something like p/q, where q is the minimum crossing number and p the minimum possible max(m,n). Do you know any previous work along this scheme (especially those using a different naming or notation, but are equivalent)?


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