Spectral gap and dispersion of the knot scheme

Determine the spectral gap of the knot scheme's move graph and establish whether its Alexander-polynomial serial numbers are dispersed.

Background

The paper identifies rapid mixing, dispersed serial numbers, and large orbits as design conditions relevant to security. The graph example satisfies all three but is broken, while the paper does not establish these conditions for the knot scheme.

For the knot scheme of Farhi et al., the spectral gap of the move graph controls whether verification rapidly projects onto orbit states, and dispersion of the Alexander polynomial controls whether independently minted notes are unlikely to share a serial number. Both properties are explicitly left unresolved.

References

For the knot scheme, the spectral gap of the move graph and the dispersion of the Alexander polynomial are both open.

— Path-Finding, Orbit State Preparation, and the Security of Invariant Quantum Money  (2609.39774 - Schmiedel et al., 30 Sep 2026) in Section 10, Conclusion, paragraph “Does an unbroken scheme meet the design conditions?”