I am working on knot theory and basic algebraic topology. In order to prove that the figure eight knot and the trefoil knot are not isotopic, I have to show that their knot groups (i.e. the fundamental groups of their complements) are not isomorphic, using their Wirtinger presentation.
The groups are the following: $$G(H) = \langle a, d \mid a d a^{-1} d a = d a d^{-1} a d \rangle$$ and $$G(T_1) = \langle p, q \mid q^2 = p^3 \rangle.$$ Of course, these groups are without torsion, so I cannot use any argument about orders of elements.
How can I proceed? Thank very much.
[This is N. Owad's remark, just spelled out]
The second group affords an epimorphism to $S_3$, namely $p\mapsto (1,2,3)$, $q\mapsto(1,2)$. On the other hand, the first group affords no such epimorphism (try out all 36 possible generator images, use van Dyck's theorem).