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I am looking at this tutorial (How to make tied knots in blender?) and I want to make the curve into a watertight model for 3D printing. Is this possible?

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You can! In fact, you can also save yourself a couple of steps! You don't actually have to convert it to a curve at all. All you have to do is use the Skin modifier. This will make a cube around each vertex connected to adjacent vertices. Unlike with curves, it also supports branching. Like curves, you can control the size of each point (the vertices, in this case). This is actually a well-documented feature, so you shouldn't have any trouble finding a tutorial on the basics.

Additionally, if you want your knots smoother, you can start from a curve and convert it into a mesh (it will create vertices based on your curve resolution), then proceed to add your skin modifier.

If you already have an existing object created, you can simply convert it into a mesh, as well, then delete any extraneous curves that remain.

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