HEXAGONbeta

The bridge index and the nonorientable four-genus of knots with at most thirteen crossings

A mix of human-written and AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.GR — Group Theory

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We determine two invariants for knots that the KnotInfo database leaves open. KnotInfo lists the bridge index of \btotal{} prime knots with thirteen crossings as either 33 or 44; we prove that \bclosed{} of them have bridge index and meridional rank 44, so the meridional rank conjecture holds for them. The lower bound comes from representations of the knot group in $\GL_4(\F_q)$ that send a meridian to a transvection or a reflection and fix no nonzero vector. KnotInfo leaves the smooth nonorientable four-genus open for \ntotal{} prime knots with eleven to thirteen crossings; we determine it for \nsettled{} of them and narrow the interval for \nnarrowed{} more. Upper bounds come from pinch moves, lower bounds from the Goeritz-lattice obstructions of Jabuka and Kelly evaluated by an exhaustive search for lattice embeddings. Every bound is a certificate checked by a program independent of the search that found it.

Provenance statement

Deep Bhattacharjee set the task and directed the work. Claude (Anthropic), through Claude Code, chose the open KnotInfo entries, designed and ran the searches (SnapPy, Regina, exhaustive lattice-embedding search), wrote the certificates and the independent verifiers, and drafted and revised the paper. Every bound is a certificate that a verifier independent of the search checks: 243 bridge-index certificates (Python standard library), 2,728 pinch certificates (Regina) and 382 lattice lower bounds (recomputed with shuffled search order and tested against the recorded values of KnotInfo). Code, data and LaTeX source: https://github.com/creelie/conjecture-to-choose, release v1.3.0, archived on Zenodo, doi:10.5281/zenodo.23561157.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026