We have the following result.
Theorem K is alternating if and only if the Turaev genus of K is 0.
Lowrance has proved in On knot Floer width and Turaev genus that the Turaev genus is an upper bound for the width of the Heegaard Floer knot homology, minus 1. A similar bound for the Khovanov width was found by Manturov in Minimal diagrams of classical and virtual links. See also Champanerkar, Kofman and Stoltzfus.
In [DFKLS] it is proved that the Turaev genus is bounded above by the crossing number minus the span of Jones polynomial.
There is a related invariant. Every knot K is isotopic to an embedding into a regular neighborhood of a standardly embedded surface of genus g in S3, Fg. If g is large enough, there exists such an embedding which is alternating with respect to the height function on the regular neighborhood, given by projecting on the I factor in the neighborhood, Fg x I. The minimum genus g for which such an embedding exists might be called the alternating genus of K. This provides a lower bound for the Turaev genus.
Abe and Kishimoto have shown in The dealternating number and the alternation number of a closed 3-braid, Corollary 5.5, that the Turaev genus for all nonalternating knots under 12 crossings is 1, except for 11n_95 and 11n_118. For these two remaining knots, it might be either be 1 or 2.
[CKS] A. Champanerkar, I. Kofman, N. Stoltzfus, Graphs on sufaces and the Khovanov homology, Alg. and Geom. Top. 7 (2007) 1531-1540.
[DFKLS] O. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin, and N. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Comb. Theory, Series B, Vol 98/2, 2008, pp 384-399.
[T] V. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, Enseign. Math. (2) 33 (3-4), 203-225, 1987.