In the paper [S. Kolyada, M. Misiurewicz and L. Snoha, Spaces of transitive interval maps, Ergod. Th. & Dynam. Sys., posted electronically on August 5, 2014] we investigated topology of spaces of continuous topologically transitive interval maps. Let $T_n$ denote the space of all transitive maps of modality $n$. For every $n>0$, in the union of $T_n$ and $T_{n+1}$ we constructed a loop (call it $L_n$), which is not contractible in in this space. One of the main open problems left in that paper was whether $L_n$ can be contracted in the union of more of spaces $T_i$.

Since all elements of the loops $L_n$ are maps of constant slope, we are studying the spaces $TCS_n$ of transitive maps of modality $n$ and constant slope. We show that for every $n>1$ the loops $L_n$ and $L_{n+1}$ can be contracted in the union of $TCS_n$, $TCS_{n+1}$ and $TCS_{n+2}$. Moreover, the loop $L_1$ can be contracted in the union of $TCS_1$, $TCS_2$ and $TCS_4$.

Additionally, we describe completely the topology (and geometry, in a certain parametrization) of the spaces $TCS_1$, $TCS_2$ and their union. In particular, we show that this union is homotopically equivalent to the circle.