Here is the script: BraidWitt.sage.

Quick instructions:

sage: load "BraidWitt.sage"
sage: b= BraidBurau( [(1, 3),  (2, -1)], 4 )
sage: pl= b.plot(); show(pl, axes= False, figsize= (8,2))
sage: b.det()
sage: P= b.histogram(); show(P)

Basic usage in more details:

-- Start Sage.

-- Enter:

sage: load "BraidWitt.sage"

This assumes that the file you are reading now is in your Sage path. If not, enter:

sage: load "path/to/your/BraidWitt.sage"

instead.

-- Braids are to be given as lists of pairs (i,j), each representing sigma_i^j. So for example [(1, 3), (2, -1)] represents sigma_1^3sigma_2^(-1). Of course this may be seen in B_3 or in B_n for any n > 2, so it will be necessary to specify n. The bottom line is that you can create a braid using 'BraidBurau', a list of pairs, and an integer n, for example:

sage: b= BraidBurau( [(1, 3),  (2, -1)], 4 )

Here b is the braid sigma_1^3sigma_2^(-1) viewed in B_4.

-- Enter:

sage: pl= b.plot(); show(pl, axes= False, figsize= (8,2))

to see a plot of the braid, draw from left to right rather than the usual top to bottom. Experiment with different values for 'figsize'.

-- Enter:

sage: b.det()

to compute the signed determinant, which is the Laurent polynomial invariant defined in our paper. Note that the computation may take some time, depending on the braid. The second time you run this command though, or when you try the next commands, you should get the result at once.

-- Enter:

sage: P= b.histogram(); show(P)

to get a quick diagram of the various theta-signatures which we define. If you want to produce the 'camembert' picture, try:

sage: camembert, colours= b.camembert()

You can then see the picture by typing 'show(camembert, axes= False, aspect_ratio=1)'. To see what signatures correspond to the colours, you can either call b.histogram() and read them on this graph. Otherwise, you can inspect the variable 'colours', which contains a list of pairs (img, n) where img is a small rectangular image which can be seen with 'show(img)', allowing you to see a colour, and n is the signature for that colour, as showing on the camembert. (Try it! try 'img, n= colours[0]' first.)

-- Enter:

sage: L= b.Witt()

to get the diagonal entries of a representative for our invariant in the Witt ring. From this you can forge your own calculations beyond the determinant and signatures.

-- Also note that conway_poly(3, [(1, 3), (2, -1)], t) returns the Alexander-Conway polynomial for the same link, evaluated at t (that is, the object 't' must exist in Sage previously, for example you can use t= ZZ['t'].fraction_field().gen()).