Discrete p-Density and Compression Radii of Lattice Knots

We develop a framework for discrete p-density and compression-radius profiles of lattice knots. For lattice polygons representing a fixed knot type, we define scale-free density quantities by dividing lattice length by chord-length spread functionals, and we define corresponding compression radii using a raw lattice thickness convention. These profiles are studied both on length-filtered lattice representative sets and on seed-generated finite search spaces arising from local move systems such as BFACF moves. The framework is intended as a computable counterpart of continuous scale-free geometric invariants of knot types. It separates density, compression, and ropelength-like quantities, and records how these quantities behave under length filtration and finite move-graph exploration. We emphasize throughout the distinction between exhaustive lattice-knot invariants and seed-generated computational profiles. As a case study, we compute raw p-density and compression-radius values along an explicit BFACF merge certificate connecting a 30-edge simple cubic figure-eight seed to its reflected mirror seed at length bound N=32. The extracted path has 21 states and 20 BFACF moves and passes through a 32-edge connecting state. Along this path, density and compression-radius values are not monotone: the connecting state has larger raw density and slightly larger D_2-compression radius than the seeds, while an intermediate 30-edge state has smaller compression radius but larger density. This illustrates that density minimization, compression minimization, and ropelength-like minimization define distinct finite-state optimization problems.