Examples of isomorphic group presentations with homotopy NONequivalent complexes 
(Wikipedia) A presentation complex is a 2-dimensional cell complex associated to any presentation of a group $G$. The complex has a single vertex, and one loop at the vertex for each generator of $G$. There is one 2-cell for each relation in the presentation, with the boundary of the 2-cell attached along the appropriate word.

A group $G$ can be presented in different ways. These presentations need not have homotopy equivalent presentation complexes. I would like to collect as many examples of presentations of a group with presentation complexes that are not homotopy equivalent as possible. I would take the following as sufficient answers to this post:

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*Referrals to references which have such examples, or discuss methods to construct such presentations;

*Examples of such presentations, preferably a large number of them, and preferably not all sharing many common threads (in, say, how they are constructed, dimension, etc.).

The motivation for my question is that I am interested in so-called exotic group presentations. See here for some info on them.
 A: Take
$$G=\langle x,y\mid x^2=y^3\rangle.$$
This is the standard presentation for the fundamental group of the trefoil knot. Of note is the fact that this group is Hopfian and non polycyclic-by-finite.
Now for each $n\in\mathbb{N}$ put
$$G_n=\langle x,y,\overline x,\overline y\mid x^2=y^3,\overline x^2=\overline y^3,x^{2n+1}=\overline x^{2n+1},y^{3n+1}=\overline y^{3n+1}\rangle.$$
The result in each case is a different presentation of $G$. Write $K_n$ for the presentation complex corresponding to $G_n$. Using the fact that the commutator group of $\pi_1K_n\cong G_n\cong G$ is free of rank $2$, and that $G/[G,G]\cong\mathbb{Z}$ you can show without too much difficulty that that $\chi(K_n)=1$ for each $n$.

There are infinitely many distinct homotopy types amongst the $K_n$.

This may be established by studying the homologies of the universal covering spaces $\widetilde K_n$. It can be shown that there are infinitely many integers $m,n$ such that $H_2(\widetilde K_m;\mathbb{Z})$ and $H_2(\widetilde K_n;\mathbb{Z})$ are non-isomorphic $\mathbb{Z}[G]$-modules. An interesting fact is that $K_m\vee S^2\simeq K_n\vee S^2$ although $K_m\not\simeq K_n$.
The calculations for all this are contained in Exotic relation modules and homotopy types for certain 1–relator groups, Algr. Geo. Top. 6, 2163-2173, (2006), by J. Harlander, J. Jensen. They make use of ealier work by P. Berridge and M. Dunwoody in the paper Non‐Free Projective Modules for Torsion‐Free Groups J. London Math. Soc. 2, 433–436, (1979).
