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 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.

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 In unpublished work (May 14, 2014), Joshua Howie has shown that of these, all have Turaev genus at most 2.

Specific Knots

11n95, 12n253, 12n254, 12n280, 12n323, 12n356, 12n375, 12n452, 12n706729, 12n811, 12n873
   Oliver Dasbach and Adam Lowrance have computed that the Turaev genus of each of these knots is 2. See [4].

References

[1] Abe, T. and Kishimoto, K., "The dealternating number and the alternation number of a closed 3-braid," Arxiv preprint.

[2] Champanerkar, A., Kofman, I., and Stoltzfus, N., "Graphs on sufaces and the Khovanov homology," Alg. and Geom. Top. 7 (2007), 1531-1540.

[3] Dasbach, O., Futer, D., Kalfagianni, E., Lin, X-S., and Stoltzfus, N., "The Jones polynomial and graphs on surfaces," J. Comb. Theory, Series B, Vol 98/2 (2008), 384-399.

[4] Dasbach, O. and Lowrance, A., "Invariants for Turaev genus one links," Arxiv preprint.

[5] Jablan, S., "Almost alternating knots with 12 crossings and Turaev genus," Arxiv preprint

[6] Lowrance, A., "On knot Floer width and Turaev genus," Arxiv preprint.

[7] Manturov, V. O., "Minimal diagrams of classical and virtual links," Arxiv preprint.

[8] Turaev, V., "A simple proof of the Murasugi and Kauffman theorems on alternating links," New Developments in the Theory of Knots (1990), 602-624.