Perturbed Alexander invariants via quantum cluster algebras

Mathematica program to derive a perturbative expansion of knot invariants

© 2026 Boudewijn Bosch

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Acknowledgements

Boudewijn Bosch acknowledges support by the Dutch Research Council (NWO) under grant number 613.009.152.

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Table of Contents

• What is this program?
• Getting Started
  • Installing NCAlgebra
• Usage
• Example Execution
• Contributors

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What is this program?

Using this program, a perturbed R-matrix is derived using the theory of quantum cluster algebras. The perturbed R-matrix can then be used to derive perturbed Alexander polynomials using the perturbed Gaussian formalism provided in the Mathematica code by Roland van der Veen and Dror Bar-Natan, available at https://www.rolandvdv.nl/PG/. The input is a Gauss code for a knot diagram, and the output is a perturbed Alexander invariant.

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Getting Started

This package requires Mathematica along with the following dependencies:

• NCAlgebra — a noncommutative algebra package for Mathematica
• NCGBX — the noncommutative Gröbner basis package (ships with NCAlgebra)

Installing NCAlgebra

Starting with version 6, NCAlgebra is installed via its paclet distribution. In Mathematica, run:

PacletInstall["https://github.com/NCAlgebra/NC/blob/master/NCAlgebra-6.0.3.paclet?raw=true"]

Once installed, load NCAlgebra and NCGBX with:

<< NC`
<< NCAlgebra`
<< NCGBX`

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Usage

The function \text{Pert}_{i,j} computes the perturbed R-matrix. This is automatically computed by evaluating all initialization cells.

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Example Execution

Perturbed Alexander polynomials for the trefoil:

Z31 = hR15 hR62 hR37 C4 hv8 hv9 hv10;
Do[Z31 = Z31 //. hm[i, 1] -> 1, {i, 2, 10}]

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Contributors

Boudewijn Bosch