I've been messing around in SolidWorks for the past week or so, familiarizing myself with the process of twisty puzzle creation, when I came across this minor issue. In the attached image you can see a sphere core that is in the midst of production, but there are extra lines that have formed on the sphere after the revolve-pattern-split process. I can select the entire sphere by selecting it as a solid body, but their existence is bothering me! Does anyone know how I can get rid of them so the sphere is whole? I've tried using Knit, but it doesn't do exactly what I'm looking for.

Is the sphere now a single solid body or does it consist of several parts (whose outlines you actually see)? Do you want to combine the parts into a single solid? Use Combine feature.

Yeah, this is a known issue and I have a theory about what causes it: when you cut the puzzle all at once they don't appear, but when you minimize load on your computer by first cutting away portions of the solid, these edges appear. Now I feel strongly opposed to cutting a puzzle all at once (you get RSI from checking all the boxes and your file takes ages to rebuild) so I don't think you can avoid creating these ghost edges.
Knit is for connecting surface bodies to form a new surface or solid body. However, I just (while writing this post) discovered a clever way to fix this: you can use face delete on the offending faces, just select all but one of them and the resulting body should be nice again.

I have never made a puzzle that didn't have this problem, and it happens on more than just the core. It will occur on internal parts as well, generally if there is multiple shells. It doesn't effect the model, it will not print with the lines there, and it behaves as solid geometry. I wouldn't worry about it. It is bothering you that much delete the sphere and make a new one the same size. quick and simple fix.

see simple Video (sorry I use russian SW)
http://www.youtube.com/watch?v=vKSJpTYSoIk

I am also familiar with this phenomenon, and the only place where it gives me problems is when I need to make reference geometry for doing the finishing steps on the core (though sometimes they are even useful for the clean-up). And I have developed an even simpler fix than Tom's.

I simply make a revolved cut around the sphere, with a sketch like in the mock-up below. I always make the core of my puzzles between 0.5mm and 1mm smaller than the mechanism of the pieces that sit above it. This technique takes care of the extra lines and adds in the extra space at the same time. Two birds, one stone.

On some puzzles, I've actually taken advantage of the ghost edges, usually when the geometry of the puzzle is insane (Pentagonal Dipyramid, Deltoidal Icositetrahedron, etc.). I can use those ghost edges to create reference geometry (points, axes, planes) rather than trying to work out exact angles or dimensions, which would suck.

-Eitan