Merge Trees of Lattice Knots

Abstract

We study length-filtered move graphs of lattice knots as finite-state models for quantitative reconfiguration. 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. The first level at which two components merge defines a discrete merge scale and, after subtracting the birth level, an ultrapseudometric on eventually merging components. We specialize to the simple cubic lattice with standard BFACF moves, using classical BFACF ergodicity while distinguishing global connectivity from connectivity under a fixed length cap. We completely determine the minimal-layer BFACF merge trees of the amphichiral knots $4_1$ and $6_3$, without identifying reflections. For $4_1$, the 152 minimal states at length 30 form four components of sizes $58,58,18,18$, and all merge at $N=32$. For $6_3$, the 148 minimal states at length 40 form twelve components of sizes $39,39,10,10,9,9,7,7,6,6,3,3$. At $N=42$ these merge into two disjoint mirror-related components, each containing 74 minimal states and 12337 bounded states; the two branches merge at $N=44$. Thus the complete merge trees are $4\to1$ and $12\to2\to1$, with excess-length barriers $0,2$ and $0,2,4$, respectively. Explicit BFACF paths provide independently verifiable certificates for the extremal merges.