What Can Knitting Machines Make?

Industrial knitting machines are used to fabricate many complex 3D objects, including gloves, sweaters, and shaped fiber reinforcement for composites. But what are the limits of these machines? This talk will introduce two of the approaches my group has taken to answering this question.
The first -- top-down -- approach views a knitting machine as a device for shaping, splitting, and merging tubes. This leads to the observation that all manifolds with boundary whose Reeb graph under some smooth function has an upward-planar embedding are machine-knittable. This observation forms the basis of our work on knitting design tools.
The second -- bottom-up -- approach translates low-level machine instructions into a geometric description of intertwined yarns, allowing direct enumeration of structures. The important insight made in this work is that the way the yarn is routed between stitches is as important to model as the way the yarn behaves within stitches.
Taken together, these two views of knitting machine capabilities have enabled us to develop new design and simulation tools for machine knitting.