The Rasmussen invariant of a homogeneous knot

Abstract

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.

An estimate of the Rasmussen invariant for links and the determination for certain links

“…which Rudolph announced in [17], and completely proved that it holds for any link L and any diagram D L of L. By an argument similar to that for the proof of this inequality, we improve the above inequalities (1) and (3) as follows. For a link L and a diagram D L of L, we eliminate all negative crossings in the same manner as the Seifert algorithm, and denote the obtained diagram by D 0+…”

“…By Theorem 1.5, Remark 6.3. In [1], Abe has shown that the equality of the inequality in Theorem 1.5 holds for homogeneous knot diagrams. He had informed the author that this equality holds for connected homogeneous link diagrams.…”

A Bennequin-Type Inequality and Combinatorial Bounds

“…of (Cbound) and of (KwC15) in the case of the diagram K h in Figure 3. Notice that K h is a homogeneous diagram in the sense of [1], and the corresponding knot type does not depend on h. Theorem 16 ([27]). Let D be an almost-positive diagram of a non-split link L with a negative crossing p.…”

“…It is possible to associate to D a graph Γ(D), which we call the simplified Seifert graph. The vertices of Γ(D) are the circles in the oriented resolution of D, and there is an edge between two vertices if the corresponding circles shared at least a crossing in D. The edges of the simplified Seifert graph can be divided into three classes: (1) positive, if the circles corresponding to the endpoints of the edge shared only positive crossings, (2) negative if the circles corresponding to the endpoints of the edge shared only negative crossings, and (3) neutral, if the edge is neither positive nor negative. A vertex v of Γ(D) is called positive (resp.…”

Homogeneous links and the Seifert matrix

“…The skein relation is then considered, at both the level of the diagram and the corresponding Seifert graph, having in mind that to obtain terms involving powers of z when resolving the resolution tree, a crossing must be smoothed in the diagram D, or equivalently, an edge must be deleted from the graph G. A direct proof of the corollary has been recent and independently suggested by M. Hirasawa. The proof, outlined in a paper by Tetsuya Abe [1] (see also [17]), is strongly based on a difficult result by Gabai [8], which states that the sum of Murasugi of minimal genus surfaces is a minimal genus surface. Hirasawa applies this result to the portions F i above defined.…”