Diagonal knots and the tau invariant

TL;DR

This paper explores the properties of knots with diagonal grid diagram representations, proving they are positive knots and providing an example of a non-torus knot with such a representation, expanding understanding of the tau invariant.

Contribution

It demonstrates that all knots with diagonal grid diagrams are positive and presents the first example of a non-torus knot with such a diagram.

Findings

All diagonal grid diagram knots are positive.

Existence of non-torus knots with diagonal grid diagrams.

Enhanced understanding of the tau invariant for these knots.

Abstract

In 2003, Ozsv\'ath, Szab\'o, and Rasmussen introduced the $\tau$ invariant for knots, and in 2011, Sarkar published a computational shortcut for the $\tau$ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots, and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot.