Discrete Knot Theory via Lattice-Filtered Move Graphs

We introduce lattice-filtered move graphs as finite-state experimental models for knot types. At level N, vertices are lattice-polygon representatives of a fixed knot type with lattice length at most N, modulo orientation-preserving lattice isometries, and edges are prescribed local moves. Connected components of these graphs are discrete analogues of admissible components in ropelength-filtered knot spaces. The first level at which two initial components become connected defines a discrete merge scale; after subtracting the birth level, the resulting function is an ultrapseudometric whenever the relevant initial components eventually merge. The general framework is move-system independent. We then specialize to the simple cubic lattice and BFACF-type moves, treating BFACF as a chosen local move system rather than as a complete lattice-isotopy calculus. The main seed-generated computation uses a 30-edge simple cubic lattice seed for the figure-eight knot and its reflected mirror seed. With mirror symmetries not identified, the two BFACF components are separated at N=30 and merge at N=32. We also extract an explicit merge certificate: a 21-state, 20-move BFACF path through a 32-edge connecting state. Thus, relative to the supplied seeds and the BFACF move system, the seed-generated merge scale is 32. The result is seed-specific and move-system-specific, not a claim about the global merge matrix of the full minimal lattice layer.