Abstract:We introduce an experimental framework for discrete knot theory based on lattice-filtered move graphs. For a knot type K, the vertices are simple cubic lattice polygons of length at most N, and the edges are given by BFACF-type local moves. The connected components of these finite graphs provide a computable analogue of admissible components in ropelength-filtered knot spaces. The first level at which two seed-generated components merge defines a seed-generated merge scale. We prove that these merge scales define an ultrapseudometric on the set of components, relative to the chosen lattice and move system. The main computation concerns the figure-eight knot: a 30-edge simple cubic lattice representative of 4_1 and its reflected mirror seed are separated at the minimal level N=30, but merge at N=32. Thus the seed-generated BFACF merge scale for the chosen pair is 32. This is a seed-specific and BFACF-specific length-barrier computation, not a claim about the global merge matrix of the full minimal lattice layer. Additional computations for 3_1, 5_1, and small sanity checks for composite trefoils illustrate the method and its limitations. The paper also discusses connections with ropelength-filtered knot spaces, finite Reidemeister data, PL-realizable local move systems, and cubulated moves. Reproducible Python code and seed data are included as ancillary files.
| Comments: | 24 pages; supplementary Python code and seed-generated BFACF computation data archived at Zenodo DOI: https://doi.org/10.5281/zenodo.20370841 |
| Subjects: | Geometric Topology (math.GT); Combinatorics (math.CO) |
| Cite as: | arXiv:2605.25322 [math.GT] |
| (or arXiv:2605.25322v1 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25322 arXiv-issued DOI via DataCite (pending registration) |
From: Makoto Ozawa [view email]
[v1]
Mon, 25 May 2026 01:07:54 UTC (32 KB)
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