# On the Turaev genus of torus knots

## TL;DR

This paper calculates the Turaev genus and dealternating number for torus knots with up to five strands, providing exact values or tight bounds, and extends some bounds to links with six strands.

## Contribution

It offers the first precise or near-precise measurements of these invariants for small-strand torus knots and links, advancing understanding of their complexity.

## Key findings

- Exact Turaev genus for torus knots with ≤5 strands
- Dealternating number bounds for these knots
- Bounds on invariants for certain torus links with ≤6 strands

## 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.