Thanks. One more question. I would like to know what the presentation matrix of a two strand braid coloured with an alexander quandle look like? I don't get how the matrix B from corollary 5.2 is associated with the boxes.

@Robin thanks. But I don't quite understand how. Can you provide more explanation?

Thanks. I spent some time trying to fully understand your argument, now I'll try to prove that a curve that is homologically trivial in a solid torus is null-homotopic.

I meant the first case. Thank you for you help.

Thank you for your answer! It helped me to look at the problem differently. To be more precise, I'd like to prove that if the linking number of the two components of the link is zero then the 2-link is trivial (up to link homotopy). Can you provide more help on this?