NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT

By M. Hedden. 2005

164 citations recorded by the source corpus.

In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsvath and Szabo and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive.

Subfields: Genetics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/24f43996266adb39f07e7d1c1a5bfa2fbe529fe6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Genetics

1. Liam Watson
2. Philip Ording
3. M. Hedden

In Geometry and Topology

1. Liam Watson
2. M. Hedden
3. Philip Ording
4. Jacob Rasmussen
5. Jennifer Hom
6. David Krcatovich
7. Yi Ni
8. E. Gorsky

In Mathematical Physics

1. Liam Watson
2. M. Hedden
3. Philip Ording
4. Jacob Rasmussen
5. Jennifer Hom
6. David Krcatovich
7. Yi Ni
8. E. Gorsky

Citations in the corpus

• On knot Floer homology and cabling (2004, background): “≥ 0. Then n ≥ −2 τ(K) p−1 + 1 p . ON KNOT FLOER HOMOLOGY AND CABLING II 19 This is relevant in light of the connection between τ(K) and complex curves. Suppose that K is a C-knot. Then results of [31, 13, 36] show that τ(K) = g 4(K). In particular, τ(K) ≥ 0. Combining this fact with the corollary ”
• The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, background): “This follows from work of the first author [5].”
• On the geography and botany of knot Floer homology (2014, background): “g(K ) = g4(K ) = τ(K ) = s(K ) Various authors, see [28] The coefficients of K (t) take values in {−1, 0, 1} Ozsváth and Szabó [71, Theorem 1.”