Identifying a group that originated from a Wirtinger presentation.

Question

Given the group presentation $$ \langle x_1,x_2,x_3,x_4,x_5,x_6\mid x_2x_1x_2^{-1}x_4^{-1}, x_3x_2x_3^{-1}x_5^{-1}, x_4x_5x_4^{-1}x_2^{-1}, x_1x_3x_1^{-1}x_6^{-1}, x_5x_6x_5^{-1}x_3^{-1}, x_6x_4x_6^{-1}x_1^{-1} \rangle. $$ I would like to simplify this presentation, or even identify the group. However I am stuck on this problem for a couple of days. The group originated from a Wirtinger presentation of the link $6_1^3$ (http://katlas.math.toronto.edu/wiki/L6a5). Help is really appreciated.

Answer 1

The presentation simplifies to $$\langle x_1,x_2,x_3 \mid [x_1,x_2^{-1}] = [x_2,x_3^{-1}] = [x_3,x_1^{-1}] \rangle$$ (where $[a,b] := a^{-1}b^{-1}ab$).

I cannot identify the group (whatever that means). It is an infinite automatic group that does not appear to be hyperbolic.