lcl833-master

lcl833-master computes exact Jones polynomials from braid words using the Kauffman bracket state sum, evaluates them at the fifth root of unity $q = \exp(2\pi i/5)$, and derives calibrated LCL-833 / SATI-CODEX protection metrics ($\delta_{eff}, \alpha_{op}, G_{gap}, \epsilon_{eff}, \omega, T_{min}$) directly in your terminal or Python environment.

Features

• Exact Jones Polynomials: Uses fractions.Fraction for rational exponents to maintain exactness during intermediate steps.
• Parallel Execution: Multiprocessing support for complex braids (>= 12 crossings) to significantly reduce computation time.
• LaTeX Export: Generate publication-ready TeX strings for any computed polynomial.
• Expanded Presets: Full support for standard knots from the Rolfsen table (3_1 through 7_1).
• LCL-833 Metrics: Calibrated against the trefoil knot anchor ($|J(3_1; q_5)| \approx 1.543$).
• CLI & API: Use it as a standalone tool or a Python library.

Installation

pip install lcl833-master

CLI Usage

Basic Commands

# Run a preset knot (3_1, 4_1, figure8, etc.)
lcl833 knot 3_1

# Run a custom braid word (Artin generators σ_i)
lcl833 braid 1 1 1
lcl833 braid 1 -2 1 -2

# Emit JSON output (includes LaTeX strings)
lcl833 knot 3_1 --json

Advanced Options

• --table: Show a comparison table of all preset knots.
• --genus: Logical genus (default: 5).
• --gamma: Khovanov correction factor (default: 0.05).
• --max-crossings: Maximum allowed crossings (default: 18).
• --no-lcl: Skip calibrated metrics and only show Jones results.

lcl833 table

Python API Usage

from lcl833_master import compute_jones_result, compute_lcl833_metrics

# Compute Jones polynomial for the trefoil (σ₁³)
res = compute_jones_result((1, 1, 1))
print(f"Jones Polynomial: {res.normalized_jones_pretty}")
print(f"LaTeX: {res.normalized_jones_latex}")
print(f"Magnitude at q5: {res.magnitude_at_q5:.4f}")

# Compute LCL-833 metrics
metrics = compute_lcl833_metrics(v_j=res.magnitude_at_q5, genus=5)
print(f"Alpha_op: {metrics.alpha_op:.4f}")
print(f"T_min: {metrics.t_min}")

Performance Note

Kauffman bracket computation is $O(2^n)$. For braids with $n \ge 12$, the library automatically utilizes all available CPU cores via multiprocessing. Use --max-crossings to prevent accidental long-running processes on extremely large diagrams.

License

MIT License. See LICENSE for details.