Pearl necklace knots with fewer vertices than an equivalent FCC lattice knot

Abstract: The pearl necklace number, $N_P$, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice $N_L$, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A LLM was prompted to find configurations of knot for which $N_P<N_L$, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with $N_P=N_L-1$ were found for all knots from 5 to 7 crossings as well as $8_{19}$, and $10_{124}$. One example, $8_1$, could be constructed with $N_P=N_L-2$.

• The paper demonstrates that pearl necklace numbers can be smaller than minimal FCC lattice embedding sizes, with every tested prime knot through seven crossings improving by one sphere and the 8₁ knot by two.
• The authors use topology-preserving numerical relaxation, clearance checks, exact edge-length refinement, and an algebraic parameterization to validate configurations such as 21 spheres for 5₁ and 32 for 8₁.
• The results challenge the conjecture that pearl necklace and FCC lattice counts always agree, while leaving the 14-sphere trefoil, growth of the gap, and a general constructive mechanism unresolved.

Background and problem statement

A pearl necklace knot is a closed chain of congruent unit spheres in which each sphere is tangent to exactly its two neighbors and no non-adjacent pair overlaps. The pearl necklace number $N_P$ of a knot type is the minimum number of spheres admitting such a construction. Prior to this work, the best available bounds were the stick number from below and the minimum FCC lattice embedding size $N_L$ from above, with $N_L$ tabulated up to 10 crossings by Janse van Rensburg and Rechnitzer [(Zhang et al., 2011)-style reference, van2011minimal]. Because every FCC lattice polygon can be thickened into a pearl necklace with one sphere per lattice site, it was natural to conjecture that $N_P = N_L$, which would imply a 15-sphere lower bound for the trefoil. This question was posed explicitly in a recent open-problems survey in low-dimensional topology (Baykur–Kirby–Ruberman). The paper reports the first configurations demonstrating that the conjecture fails: there exist knots with $N_P < N_L$.

Methodology

The configurations were obtained by prompting ChatGPT (GPT-5.6 Pro) with target vertex counts and seed data drawn from Andrew Rechnitzer's catalog of minimal FCC lattice knots. The LLM's approach reduces to removing a vertex from an FCC embedding and relaxing the resulting open chain toward equilateral geometry under topology-preserving constraints—effectively constrained nonlinear optimization, apparently gradient-descent-based, supported by substantial computation. The author supplied domain heuristics: non-alternating torus knots were targeted first because their ropelength scaling suggests they need the fewest spheres per crossing.

Validation proceeded on three fronts:

• Topology: Alexander polynomial evaluation confirmed the knot type was preserved through relaxation.
• Non-overlap: minimum clearance (the smallest excess distance above 2 between non-adjacent spheres) was checked directly; clearances ranged from roughly 0.0007 to 0.03.
• Equilateral closure: raw LLM output had edge-length deviations around $10^{-13}$, well above float precision. A human-devised refinement procedure redistributes all length error into the final edge, projects the last vertex onto the bisecting plane, and repositions it at height $h=\sqrt{4-|v|^2/4}$, yielding edges exact to within $2^{-52}$. For the $8_1$ example, ten of thirty-two edges deviate by exactly one or two units in the last bit.

Additionally, the LLM produced an exact algebraic parameterization: each edge vector is written as a harmonic function of rational multiples of $\pi$ with numerator/denominator integers of order $N_L$0, with a closure construction for the final two edges. This converts the numerical evidence for the $N_L$1 knot into an exact certificate, addressing the concern that floating-point imprecision could invalidate the constructions.

Results

The headline findings are:

Knot	$N_L$2	Best $N_L$3	Minimum clearance
$N_L$4	22	21	0.00074
$N_L$5	23	22	0.02998
$N_L$6–$N_L$7	27–28	26–27	0.0099–0.0300
$N_L$8–$N_L$9	29–31	28–30	0.0099–0.0300
$N_L$0	28	27	0.0204
$N_L$1	33	32	0.0124
$N_L$2	34	32	0.0143

Every prime knot through seven crossings admits a configuration with $N_L$3, as do the non-alternating torus knots $N_L$4 and $N_L$5. The twist knot $N_L$6 achieves $N_L$7, the largest deviation found; the author identifies this as the most significant single result. The search over 8–10 crossing knots was explicitly non-exhaustive (a subset including $N_L$8 and ten 10-crossing knots), so further examples likely exist. Attempts at a 14-sphere trefoil failed repeatedly, converging to a configuration whose minimum non-adjacent separation is 1.98579 — below the required 2 — leaving the trefoil question open.

The paper also reports semi-lattice configurations in which most vertices remain on FCC sites while a perturbed minority sits off-lattice: 16 of 26 sites for $N_L$9, 26 of 33 for $N_P = N_L$0, and 22 of 32 for $N_P = N_L$1. Structurally, the twisted regions of twist knots stay on-lattice while the clasp is displaced. This supports a picture in which surface vertices of sufficiently large lattice knots can be perturbed and removed while bulk vertices cannot.

Discussion and scaling hypotheses

Two caveats bear on whether these results truly establish $N_P = N_L$2. First, numerical imprecision could mean the configurations are not genuinely equilateral; the refinement procedure and exact harmonic parameterization make this unlikely. Second, the published $N_P = N_L$3 values could be overestimates; the author considers this improbable for knots up to eight crossings given the thoroughness of the BCACF sampling (billions of moves), though it cannot be excluded at higher crossing numbers. An exhaustive check of all self-avoiding polygons of length 21 would settle the $N_P = N_L$4 case but is computationally prohibitive.

Assuming $N_P = N_L$5 scales with crossing number $N_P = N_L$6 as ropelength does ($N_P = N_L$7 alternating, $N_P = N_L$8 non-alternating) and that only surface sites ($N_P = N_L$9 of them) are removable, the paper hypothesizes without supporting data that

$N_P < N_L$0

Testing this requires extending $N_P < N_L$1 computations beyond 10 crossings, which currently bounds all such work.

An intriguing observation connects these results to MD-energy minimization: the vertex count minimizing the minimum-distance energy appears close to both $N_P < N_L$2 and $N_P < N_L$3 (e.g., the 21-sphere $N_P < N_L$4 has MD energy near 144, minimizable to 133). This suggests a possible proof strategy: derive MD energy minima for equilateral knots and compare against global energy minima at fixed vertex count.

Limitations and open questions

Several limitations are stated plainly. The LLM-generated replication scripts did not independently reproduce the results: a Python version was too slow to run, and a MATLAB version failed to produce non-overlapping configurations during the author's tests. The crankshaft-move strategy—perturbing an FCC knot until three consecutive vertices form an equilateral triangle amenable to vertex deletion—was attempted by both human and LLM without success, so no constructive combinatorial mechanism for the off-lattice improvement is known. The reported clearances near 0.0299 recur suspiciously across many knots; the author notes this may reflect optimization stopping criteria rather than a geometric constraint, and the question is left unresolved. The 14-sphere trefoil remains open, with repeated searches converging to a sub-unit-clearance near-miss that may indicate a genuine obstruction. Finally, the claimed strengthening of Buck's ropelength constant to yield a bound $N_P < N_L$5 was not rigorously evaluated and is included only for interested readers.

Conclusion

This paper establishes, via LLM-generated and human-refined configurations, that the pearl necklace number can be strictly smaller than the minimal FCC lattice embedding size, refuting the presumed equality for all tested prime knots through seven crossings and achieving a two-vertex improvement for $N_P < N_L$6. The constructions are validated topologically, numerically, and—in one case—exactly via rational-angle trigonometric parameterization. Whether the gap grows systematically with crossing number, whether the trefoil admits 14 spheres, and whether contact-graph or MD-energy arguments can convert these numerical observations into proofs remain open.