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, password: qdesing)
- lincomb.twist.zip (linear combinations to obtain those recurrences, 29 MB, password: qdesing)
- 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, password: qdesing)
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