Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$. Then $M=M_p\cup_i M_q$ where the isomorphism $i$ identify $S^1\times \{1\}$ in $M_p$ and $M_q$.
Does $M$ have a name ? Is there some literature on this object, studying homotopy and homology groups ? Same question if we change $p$ and $q$ by arbitrary integers.
These spaces $M_{p,q}=M_p\cup M_q$ are discussed in Examples 1.24 and 1.35 of my algebraic topology book. I don't know that they have a standard name, apart from $M_{2,2}$ which is the Klein bottle. The universal cover of $M_{p,q}$ can be described explicitly as the product of an infinite tree with $\mathbb R$, as shown in Example 1.35, so the universal cover is contractible. Hence all the groups $\pi_nM_{p,q}$ are zero for $n>1$, making $M_{p,q}$ an Eilenberg-MacLane spaces $K(\pi,1)$. By van Kampen's Theorem $\pi_1M_{p,q} =\langle a,b \ |\ a^p=b^q \rangle$. The only nontrivial homology group $H_nM_{p,q}$ for $n>0$ is $H_1$ which is ${\mathbb Z} \times{\mathbb Z}_d$ for $d$ the greatest common divisor of $p$ and $q$.
When $p$ and $q$ are relatively prime, $M_{p,q}$ is a deformation retract of the complement of the $(p,q)$ torus knot in $S^3$. For example the space $M_{2,3}$ in the original question is a deformation retract of the complement of the trefoil knot. If $p$ and $q$ are not relatively prime then $M_{p,q}$ cannot be embedded in ${\mathbb R}^3$, generalizing the fact that the Klein bottle does not embed in ${\mathbb R}^3$.
