Exact loop densities in $O(1)$ dense loop model on the cylinder of odd circumference and clusters in half-turn self-dual critical percolation

We consider $O(1)$ dense loop model in a square lattice wrapped on a cylinder of odd circumference $L$ and calculate the exact densities of loops. These densities of loops are equal to the densities of critical bond percolation clusters on a forty-five-degree rotated square lattice rolled into a cylinder with special boundary conditions which we refer to as half-turn self-dual percolation. The solution is based on a correspondence between the $O(1)$ dense loop model and the six-vertex model at the Razumov-Stroganov point.