Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsvath and Z. Szabo in the early 2000's which associates to a knot K a chain complex called $CFK^{\infty}(K)$, and improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu, which additionally considers a chain map on $CFK^{\infty}(K)$ called $\iota_K$, and extracts from this data two new numerical invariants of knot concordance. These new invariants are interesting, because, unlike other concordance invariants from Heegaard Floer homology, they do not necessarily vanish on knots of finite order in the group of concordance classes of knots. The map $\iota_K$ is in general difficult to compute, and computations have been carried out for relatively few knots. We discuss computations of $iota_K$ for 10 and 11-crossing knots satisfying a certain simplicity condition, called the (1,1)-knots. Our methods are principally focused in homological algebra.