## Desingularization in the q-Weyl algebra

Christoph Koutschan and Yi Zhang

### Abstract

In this paper, we study the desingularization problem in the first q-Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q-Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q-Weyl closure of a given q-difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator.### Paper

Here is the full text (preprint) of our paper Desingularization in the q-Weyl algebra.### Supplementary electronic material

- BimonicRecurrences.m (Mathematica code for guessing bimonic recurrences)
- BimonicRecurrences.nb (Mathematica demo notebook)
- BimonicRecurrences.cdf (the notebook in cdf format for the Wolfram CDF Player)
- BimonicRecurrences.pdf (pdf screenshot of the notebook)
- The implementation of the desingularization algorithm can be found on Yi Zhang's webpage.

### Examples: twist knots

- Link to the q-recurrences for twist knots
- twistknots.calc.m (Mathematica script for dealing with all twist knots)
- twistknots.out (output of the above Mathematica script)
- bmrec.twist.zip (all computed bimonic recurrences for twist knots, 76 MB)
- lincomb.twist.zip (linear combinations to obtain those recurrences, 29 MB)
- In the table below we list some results for small twist knot recurrences:

p |
original recurrence | linear combination | bimonic recurrence | ||||||

ord | deg | file | ord | deg | file | ord | deg | file | |

-3 | 6 | 23 | rec.twist.knot.-3.m | 6 | 16 | lincomb.twist.-3.m | 12 | 18 | bmrec.twist.-3.m |

-2 | 4 | 15 | rec.twist.knot.-2.m | 4 | 9 | lincomb.twist.-2.m | 8 | 11 | bmrec.twist.-2.m |

-1 | 2 | 7 | rec.twist.knot.-1.m | 2 | 3 | lincomb.twist.-1.m | 4 | 5 | bmrec.twist.-1.m |

2 | 3 | 12 | rec.twist.knot.2.m | 2 | 4 | lincomb.twist.2.m | 5 | 9 | bmrec.twist.2.m |

3 | 5 | 20 | rec.twist.knot.3.m | 4 | 10 | lincomb.twist.3.m | 9 | 15 | bmrec.twist.3.m |

4 | 7 | 28 | rec.twist.knot.4.m | 6 | 16 | lincomb.twist.4.m | 13 | 21 | bmrec.twist.4.m |

### Examples: pretzel knots

- Link to the q-recurrences for pretzel knots
- lincomb.bmrec.pretz.zip (computed bimonic recurrences and corresponding linear combinations, 58 MB)

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