The Rasmussen invariant of a homogeneous knot

TL;DR

This paper computes the Rasmussen invariant for homogeneous knots, a class extending alternating and positive knots, providing explicit diagram-based descriptions and characterizations of positive knots.

Contribution

It introduces a method to explicitly determine the Rasmussen invariant for homogeneous knots, linking it to their diagrams and characterizing positive knots.

Findings

Rasmussen invariant explicitly described for homogeneous knots

Characterizations of positive knots derived from the invariant

Recovery of Baader's theorem relating positivity, homogeneity, and quasipositivity

Abstract

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader's theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.