An Artin Braid Group Representation of Knitting Machine State with Applications to Validation and Optimization of Fabrication Plans

Industrial knitting machines create fabric by manipulating loops held on hundreds of needles. A core problem in pattern making for these machines is transfer planning - coming up with a sequence of low-level operations that move loops to the appropriate needles so that knitting through those loops produces the correct final structure. Since each loop is connected to the larger piece in progress, transfer plans must account for not only loop position, but the way strands of yarn tangle around each other. We present the first complete, discrete representation of the machine's loop-tangling process. Our representation combines a braid from the Artin Braid Group with an array of explicit loop positions to fully capture loop crossings. By storing braids in the Symmetric Normal Form, states can be quickly compared and updated incrementally with machine operations. This representation can be used to verify the equivalence of transfer operations, providing an important tool in optimizing knit manufacturing. We improve on prior work in transfer planning algorithms, which can only solve certain subclasses of problems and are frequently suboptimal in terms of fabrication time, by introducing a novel A* search heuristic and state-collapsing mechanism, which we show finds optimal transfer plans for a large benchmark set of small transfer planning problems.