From one generator to loop order on the three-cube
Abstract
We study the monoid 𝒲 of based closed walks on the $1$-skeleton of the three-dimensional cube Q3. The fundamental group of this graph is the free group F5. We prove that every additive one-step reading, given by a sum of edge weights in an abelian group, factors through the directed transition counts, and we exhibit two closed walks with equal transition counts and different reduced loop words. Hence commutative aggregation does not determine the reduced loop word. The reduced loop word gives a proper noncommutative recognition quotient. We also determine the shortest closed walk with trivial abelianization but nontrivial degree-two commutator information. Its minimum length in the cube edge metric is $14$. We also obtain a separation between finite and unbounded memory. A two-state reading separates an order pair, while for every k≥1, there is an explicit factorial pair which no k-state reading separates. An unbounded stack recovers the reduced loop word on every walk. For integer-valued functions on the vertex set, potential readings vanish on closed walks, while occupation binding is not rectangular. More generally, every occupation-based constraint is rectangular on a class of histories if and only if the occupation vector is constant on that class. Finally, a declared quarter-turn quaternion clock gives a second proper order-sensitive congruence, incomparable with the reduced-word quotient. The obstruction to commutative recovery, and the minimum $14$, remain valid on every hypercube Qn, n≥3.
Related
- A Hadamard Cube of Order 22
- A density deficit for sums of three cube-full numbers
- Relative isoperimetry in the three-dimensional cube: classification and quantitative stability
- Fixed-Radius Neighbourhoods in Three-Dimensional Unit-Cube Graphs
- Certified computations on no-three-in-line problems: exact values and witnesses in the cube, and the Guy-Kelly count in the plane
BibTeX
@misc{washburn2026from,
title = {{From one generator to loop order on the three-cube}},
author = {Jonathan Washburn and Milan Zlatanović},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19218},
eprint = {2609.19218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19218},
}
RIS
TY - UNPB
TI - From one generator to loop order on the three-cube
AU - Jonathan Washburn
AU - Milan Zlatanović
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19218
UR - https://arxiv.org/abs/2609.19218
AB - We study the monoid 𝒲 of based closed walks on the $1$-skeleton of the three-dimensional cube Q_3. The fundamental group of this graph is the free group F_5. We prove that every additive one-step reading, given by a sum of edge weights in an abelian group, factors through the directed transition counts, and we exhibit two closed walks with equal transition counts and different reduced loop words. Hence commutative aggregation does not determine the reduced loop word. The reduced loop word gives a proper noncommutative recognition quotient. We also determine the shortest closed walk with trivial abelianization but nontrivial degree-two commutator information. Its minimum length in the cube edge metric is $14$. We also obtain a separation between finite and unbounded memory. A two-state reading separates an order pair, while for every k≥1, there is an explicit factorial pair which no k-state reading separates. An unbounded stack recovers the reduced loop word on every walk. For integer-valued functions on the vertex set, potential readings vanish on closed walks, while occupation binding is not rectangular. More generally, every occupation-based constraint is rectangular on a class of histories if and only if the occupation vector is constant on that class. Finally, a declared quarter-turn quaternion clock gives a second proper order-sensitive congruence, incomparable with the reduced-word quotient. The obstruction to commutative recovery, and the minimum $14$, remain valid on every hypercube Q_n, n≥3.
ER -