Seifert Matrices and Seifert Surfaces for the following two Brunnian links

Question

Please consider the following two Brunnian links

my questions are:

1) How to compute the Seifert Matrices for such brunnian links

2) How to obtain pictures for the corresponding Seifert surfaces.

Answer 1

"To obtain a Seifert matrix for an $n$-component Brunnian link $L$, we shall use a Seifert surface constructed by the above “algorithm”..., see page $3$ here for the construction.

Answer 2

According with a previous post Are the following two links isomorphic?, the two brunnian links were identified as the link L14n63195.

Using the following code for SnapPy

L=Link('L14n63195')
word = L.braid_word(); word

the corresponding braid was computed and the output was

[1, -2, 3, -2, 1, 3, -2, 3, -2, 1, -2, 1, -3, -2]

Now using Sage with the following code

B = BraidGroup(4)
L = Link(B([1, -2, 3, -2, 1, 3, -2, 3, -2, 1, -2, 1, -3, -2]))
L.seifert_matrix()

the Seifert matrix was computed and the result was

[-1  0  0  1  0  0  0  0  0  0  0]
[ 0  1  1 -1  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  0  0]
[ 0  0 -1  1  0  1 -1  0  0  0  0]
[ 1  0  0 -1 -1  0  0  0  1  0  0]
[ 0  0  1  0  0 -1  0  0  0  0  0]
[ 0  0  0  0  0 -1  1  1 -1  0  0]
[ 0  0  0  0  0  1  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  1  0 -1]
[ 0  0  0  0  1  0  0  0 -1 -1  1]
[ 0  0  0  0  0  0  0 -1  0  0  1]

Do you agree?

Answer 3

In order to obtain the corresponding Seifert surface, we will use the program SeifertView. We introduce the corresponding braid according with the code

AbCbACbCbAbAcb

The corresponding braid has the form

The output generated by SeifertView is the following Seifert surface

The smoothed and refined Seifert surface is