Turaev Genus


The Turaev genus of a knot was first defined in [DFKLS] below. Here is a simple definition. Every knot diagram has smoothings of type A and B. To construct the A smoothing, locally orient the arcs at each crossing so that the crossing is right handed. Smooth each crossing so that orientation is preserved. Similarly, to construct the B smoothing, smooth so that orientations are inconsistent. The two smoothings produce collections of circles in the plane, say SA and SB, with sA and sB circles, respectively. These two collections of circles are naturally cobordant via a cobordism of genus g = (2 + c - sa - sb)/2. The minimum of this genus over all diagrams for the knot is called the Turaev genus.

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.

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 In unpublished work (May 14, 2014), Joshua Howie has shown that of these, all have Turaev genus at most 2.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.