The Turaev genus of torus knots

Abstract

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strand up to an error of at most two. Additional bounds are given on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.

Extremal Khovanov homology of Turaev genus one links

“…Abe [Abe09b] proved that any adequate diagram is Turaev genus minimizing. Jin, Lowrance, Polston, and Zheng [JLPZ17] computed the Turaev genus of all 4-stranded torus knots and of many 5-stranded and 6-stranded torus knots.…”

A note on knot Floer thickness and the dealternating number

“…With Turner's calculation of the thickness of Khovanov homology for torus knots of braid index three [Tur08], Abe and Kishimoto [AK10] determine the dealternating number for all (3, q)-torus knots. For torus knots of braid index 5 or fewer, the dealternating number is computed up to an error of at most two [JLPZ17].…”