The Rasmussen invariant, four-genus and three-genus of an almost positive knot are equal

TL;DR

This paper proves that for almost positive knots, the Rasmussen invariant, 4-genus, and 3-genus are equal, extending known results from positive knots, and characterizes the Rasmussen invariant in terms of knot diagrams.

Contribution

It establishes the equality of Rasmussen invariant, 4-genus, and 3-genus for almost positive knots and determines the invariant from diagrams, also showing these knots are not homogeneous.

Findings

Rasmussen invariant, 4-genus, and 3-genus are equal for almost positive knots.

The Rasmussen invariant can be computed from the almost positive knot diagram.

No almost positive knot has 4-genus one.

Abstract

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, $4$-genus and $3$-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that any almost positive knot is not homogeneous, and there is no almost positive knot of $4$-genus one.