Fourteen-pearl trefoil conjecture

Determine whether a trefoil knot can be constructed from 14 or fewer mutually non-overlapping unit spheres arranged as a closed equilateral polygon with consecutive sphere centers separated by exactly two units.

Background

The pearl necklace number of a knot is the minimum number of unit spheres needed to form a closed equilateral polygonal knot in which consecutive spheres are tangent and non-adjacent spheres do not overlap. The FCC lattice number provides an upper bound for this quantity, and the trefoil has FCC lattice number 15.

The manuscript reports no valid 14-sphere trefoil configuration, while finding configurations with fewer spheres than the FCC lattice number for several other knots. Consequently, the question of whether the trefoil requires at least 15 spheres remains unresolved.