If L is an alternating link and c is any crossing point in a diagram, then the links formed by forming either smoothing of L, say L_0 and L_1, are both alternating and the determinants satisfy det(L) = det(L_0) + det(L_1). The set of quasi-alternating links is defined using these two essential properties.
The set of quasi-alternating links is the smallest set of links which contains the unknot and for which these two properties hold.
See [1] for 8_{19, 21}, 10_{125,126,127, 141,143, 148,149,155,157,159}
See [2] for 10_{129, 130,131, 133,134,135, 137,138,142,144,146,147,160}
See [3] for 10_{150,151,153,156,158,162,163,164,165}, 11n_50 (11n_50 is the only knot up to 11 crossings that is both KH thin and non-QA)
See [4] for 8_{19, 21}, 9_{42,43,44,45,47,48,49}
See [5] for 9_46, 10_{140}, 11n_{107,139}
[1] Baldwin, J. A., "Heegaard Floer homology and genus one, boundary component open books," Arxiv preprint.
[2] Champanerkar, A. and Kofman, I., "Twisting quasi-alternating links," Arxiv preprint.
[3] Greene, J., "A spanning tree model for the Heegaard Floer homology of a branched double-cover", Journal of Topology, 2013. Arxiv preprint.
[4] Manolescu, C., "An unoriented skein exact triangle for knot Floer homology," MRL, 2007. Arxiv preprint.
[5] Shumakovitch, A., "Patterns in odd Khovanov homology," Arxiv preprint