Galois action on homology of generalized Fermat Curves

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

Enhanced Toolpath Planning for Fused Filament Fabrication

The slicing software process the 3D geometry into 2D slices and toolpaths for additive manufacturing processes. Most slicing software allows users to select from an array of infill patterns and to specify the overall infill volume fraction globally. However, the ability to locally control the volume fraction, and mechanical properties, is often limited. In this paper, we propose a novel toolpath enhancing algorithm to enable the local control on the volume fraction of various stock and custom infill patterns. In particular, the algorithm widens the infill pattern by directly modifying their toolpath with connected Fermat curves. By preserving the topology of the original toolpath, the connected Fermat curve not only produces predictable boosts in part performance but also minimized the printing time by eliminating extruder traversals without material deposition. The field that controls local volume fraction can be designed either manually or through optimization. The effectiveness of the proposed approach in toolpath generation is demonstrated through volume fraction fields designed by both approaches.