Discrete p-Density and Compression Radii of Lattice Knots
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| Format: | Preprint |
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2026
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| _version_ | 1866910268866428928 |
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| author | Ozawa, Makoto |
| author_facet | Ozawa, Makoto |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29160 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discrete p-Density and Compression Radii of Lattice Knots Ozawa, Makoto Geometric Topology 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. |
| title | Discrete p-Density and Compression Radii of Lattice Knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2605.29160 |