Using Künneth formula I calculated $H_n(K\times S^1) = \begin{array}{cc} \{ & \begin{array}{cc} \mathbb{Z} & n= 0 \\ \mathbb{Z}^2\bigoplus \mathbb{Z}_2& n=1 \\ \mathbb{Z}\bigoplus \mathbb{Z}_2& n=2\\ 0&n\geq3 \end{array} \end{array}$
Now let $p,q\in K$ be different points in $K$. I have to calculate relative homology groups $H_n(K\times S^1,\{p,q\}\times S^1).$
It is easy to verify that $H_n(\{p,q\}\times S^1)= \begin{array}{cc} \{ & \begin{array}{cc} \mathbb{Z}^2 & n= 0 \\ \mathbb{Z}^2&n=1\\ 0&n\geq2 \end{array} \end{array}$
I'm stuck at relative long exact sequence: $$\cdots\to 0 \to \mathbb{Z}^2\bigoplus\mathbb{Z}_2 \rightarrowtail H_2(K\times S^1,\{p,q\}\times S^1) \to \mathbb{Z}^2 \twoheadrightarrow \mathbb{Z}\to H_1(K\times S^1,\{p,q\}\times S^1)\to\mathbb{Z}^2\to\mathbb{Z}\twoheadrightarrow\mathbb{Z}\to0$$
I labelled some injections and surjections, but don't know how to continue. Is there a chance to use relative Mayer-Vietoris? Thanks in advance.