Burau

Methods for finding kernel elements of the B₄ Burau representation, or helping to show that no non-trivial elements of the kernel exist.

Background

Here, we follow the lead from Bigelow¹ and consider a particular family of pairs of curves in the four-punctured disc. We follow more or less the same prescription as in Bigelow's C implementation currently available at:

https://github.com/freshbugs/burau4/blob/master/iv.c

and in particular, we thank Bigelow for useful ideas on how to represent the curves: We place the four punctures in a square, identifing them in our explanation via compass directions. In the notation of [0], we let α denote the curve connecting the two south punctures, ordered so that the code below does not have any sign mistakes:

╳ ╳
╳──╳

The curve β is more complicated. The two north punctures are placed at the middle of two caps, each consisting of a given number of parallel curves, the two south punctures at the middle of two such multi-cups, two vertical strands are pulled south from each of the two north punctures, and a multi-cap extending above the two north caps ensures that everything can be tied together.

As an example, suppose cap_west = 2, cap_east = 1, cup_west = 3, and cup_east = 2. Then, joining the north and south halves of the picture, we have:

num_strands = 2 * (cup_west + cup_east) = 10

strands, and to tie things together, we need an outer cap containing:

cap_outer = cup_west + cup_east - (cap_west + cap_east + 1) = 1

strand. At the end of the day, we get a picture that looks as follows (not including the drawing of α):

┌─────────────────┐
│ ┌───────┐ │
│ │ ┌───┐ │ ┌───┐ │
│ │ │ ╳ │ │ │ ╳ │ │
│ │ │ │ │ │ │ │ │ │
│ │ │╳│ │ │ │ │╳│ │
│ │ └─┘ │ │ │ └─┘ │
│ └─────┘ │ └─────┘
└─────────┘

Note that for this picture to make sense, we must require that:

cup_west + cup_east - (cap_west + cap_east) > 0.

Our goal is to calculate, in the notation of Bigelow², ∫_βα. As such, we follow β from the northwest puncture to the northeast puncture, along the way keeping track of intersections with α. Each intersection contributes a summand ±t^k, in which the sign of the coefficient is determined by whether we intersect from the north or the south, and whose power is determined by the current "level". To determine this power, we picture our four-punctured disk as the vertical projection of a parking garage that extends infinitely up and down (see Wikipedia³ for an illustration), with a copy of α living in each level. We start at level 0 of the garage, and as we move along, we may encounter four "ramps" that take us between different levels:

down up
──────╳ ╳──────
 up down
down up
──────╳ ╳──────
 up down

With the example β above, we first encounter an α at level 0 from the north, giving us a summand of t⁰. Then, we loop around, encounter two ramps, both of which take us down a level, before we encounter α from the north again, now at level -2, giving us a summand of t⁻². A bit later, we get a t⁻⁴ before we loop all the way around, encounter four down-ramps, then intersect α from the south, this time changing the sign, so we get a −t⁻⁸ and a bit later a −t⁻¹⁰. Adding all of these up, we find that

∫_βα = 1 + t⁻² + t⁻⁴ − t⁻⁸ − t⁻¹⁰

To show that the Burau representation of B₄ is not faithful amounts to finding a non-trivial β so that the above polynomial is 0.

Usage

The package can be installed from PyPI:

pip install burau

or it can be obtained from conda-forge:

mamba install -c conda-forge burau

The above example can be reproduced using the functionality of this Python module as follows:

>>> from burau.curve import calculate_polynomial >>> calculate_polynomial(cap_west=2, cap_east=1, cup_west=3, cup_east=2) (DictType[int64,int64]<iv=None>({0: 1, -2: 1, -4: 1, -8: -1, -10: -1}), True, 5)

Here, the first output is a dictionary mapping a power of the polynomial to the coefficient of that power. The second output indicates that the curve beta is a connected curve (an example for which this is not the case is the input (1, 1, 3, 3)). The third output is the number of crossings with α encountered along the way.

A kernel element thus corresponds to the empty dictionary. The implementation uses Numba under the hood to improve the speed of the calculation. We also provide a more vanilla Python implementation which is about 100x slower than the Numba-friendly one, but is easier to read and can be used if system restrictions make it impossible to run Numba.

References