Turaev Genus

The Turaev genus of a knot was first defined in [3]. 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. To construct the B smoothing, smooth so that orientations are inconsistent. The two smoothings produce collections of circles in the plane, say S_A and S_B, with s_A and s_B circles, respectively. These two collections of circles are naturally cobordant via a cobordism of genus g = (2 + c - s_a - s_b)/2. The minimum of this genus over all diagrams for the knot is called the Turaev genus.

Theorem: K is alternating if and only if the Turaev genus of K is 0.

Lowrance proved in [6] 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 [7]. See also [2].

In [3] it is proved that the Turaev genus is bounded above by the crossing number minus the span of Jones polynomial.

There is another invariant related to the Turaev genus. Every knot K is isotopic to an embedding into a regular neighborhood of a standardly embedded surface of genus g in S³, 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 [1], Corollary 5.5, that the Turaev genus for all nonalternating knots under 12 crossings is 1, except for 11n₉₅ and 11n₁₁₈. For these two remaining knots, it might be either be 1 or 2.

Slavek Jablan, in [5], 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 In unpublished work (May 14, 2014), Joshua Howie has shown that of these, all have Turaev genus at most 2.