\ Alexander–Gröbner 0.4.1: Getting Started

Alexander–Gröbner Computations 0.4.1

exact over QQFox calculusreduced lex Gröbner basis

This package reproduces the low-crossing calculations displayed in Section 3 of Gröbner Bases for Alexander Fitting Ideals of Links and applies the same exact pipeline to new oriented PD codes or finite presentations.

Windows: fastest route

1. Extract the Quickstart ZIP.

2. Double-click START_HERE_WINDOWS.bat.

3. Run RUN_DEMO_WINDOWS.bat to compute the bundled L8n5 example.

4. Run VERIFY_PAPER_EXAMPLES_WINDOWS.bat to recompute the frozen paper examples.

5. Drag your own JSON input onto COMPUTE_INPUT_WINDOWS.bat.

The first installation normally needs internet access to obtain SymPy 1.14.0.

PowerShell commands

py -3 -m venv .venv
.\.venv\Scripts\python.exe -m pip install alexander_grobner-0.4.1-py3-none-any.whl
.\.venv\Scripts\python.exe -m alexander_grobner doctor
.\.venv\Scripts\python.exe -m alexander_grobner demo L8n5
.\.venv\Scripts\python.exe -m alexander_grobner verify -o section3_reproduction.json

What is computed

oriented PD code / finite presentation
  -> abelianized Fox matrix A
  -> E_k = I_{n-k}(A)
  -> Laurent-to-polynomial contraction
  -> reduced lexicographic Gröbner basis AG_k

The coefficient field is QQ; the final order is t1 > ... > tell. Orientation and component order are part of the input.

Run your own PD code

.\.venv\Scripts\python.exe -m alexander_grobner compute-pd `
  --pd "X_{4,1,3,2} X_{2,3,1,4}" `
  --name Hopf -k 1 --save-input Hopf.json

Meaning of verify

verify recomputes frozen inputs and compares them with archived expected bases. It does not independently certify external table labels, decide link equivalence, or replace the mathematical proofs.

For an independent reimplementation

Read REIMPLEMENTATION_GUIDE.md. It specifies the indexing convention, PD-to-Wirtinger conversion, Fox derivative algorithm, denominator clearing, minors, saturation, final term order, and validation tests.

Documentation