The

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 S^{3}, F_{g}. 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, F_{g} 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.

Slavek Jablan, in (arxiv posting) found 154 of the nonalternating twelve crossing knots to be almost alternating, thus showing the Turaev genus is 1, leaving 37 values unknown. In unpublished work (May 14, 2014), Joshua Howie has shown that of these, all have Turaev genus at most 2.

** References**

[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.