Characterization of Positive Links and the s-invariant for Links

Abstract

We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen and Beliakova-Wehrli's s-invariant.

Positivities of Knots and Links and the Defect of Bennequin Inequality

“…Indeed, we may slide any σ 1,2 or σ 3,4 left of σ −1 1,3 using (8.3), and then the first letter of β may be assumed to be either σ 2,3 or σ 1,4 , in which case we apply (8.2). Now assume that β does not contain σ 2,4 , σ 2,3 or σ 1,4 ; that is, β = σ x 1,2 σ y 3,4 for some x, y ≥ 0. Then…”

“…Consequently, the following is a complete list of strongly quasipositive knots up to 12 crossings. Knot Strongly quasipositive braid representative Comment 3 1 1,1,1 positive braid 5 1 1,1,1,1,1 positive braid 5 2 1,1,2 (2,1,-2) 7 1 1,1,1,1,1,1,1 positive braid 7 2 1,1,(3,2,-3), (2,1,-2), 3 7 3 1,1,1,1,2, (2,1,-2) Theorem 3.1 7 4 1, (3,2,-3), (3,2,1,-2,-3), (2,1,-2),3 7 5 1, 1,1,2, (2,1,-2), (2,1,-2) Theorem 3.1 8 15 1, (3,2,-3), (3,2,-3), (3,2,1,-2,-3), (3,2,1,-2,-3),2,3 9 1 1,1,1,1,1,1,1,1,1 positive braid 9 2 2, (2,1,-2), (4,3,2,-3,-4), 2, (3,2,1,-2,-3), 4 9 3 1,1,1,1,1,1,2, (-1,2,1) Theorem 3.1 9 4 1,1,1,1, (3,2,-3), (2,1,-2), 3 9 5 2, (2,1,-2), (2,1,-2), (4,3,2,-3,-4), (3,2,1,-2,-3), 4 9 6 1,1,1,1,1, 2, (2,1,-2), (2,1,-2) Theorem 3.1 9 7 1,1,1,(3,2,-3), (2,1,-2), 3,3 9 9 1,1,1,1,2,(2,1,-2), (2,1,-2), (2,1,-2) Theorem 3.1 9 10 1, (3,2,-3), (3,2,1,-2,-3),(3,2,1,-2,-3),(3,2,1,-2,-3),(2,1,-2),3 9 13 1,1,1,(3,2,-3),(3,2,1,-2,-3),(2,1,-2),3 9 16 1,1,1,2,2,(2,1,-2),(2,1,-2),(2,1,-2) Theorem 3.1 9 18 1,1,(3,2,-3),(3,2,1,-2,-3),(3,2,1,-2,-3),(2,1,-2),3 9 23 1,1,(3,2,-3),(3,2,1,-2,-3),(2,1,-2),3,3 9 35 2, (2,1,-2), (3,2,-3), (2,1,-2), (4,3,2,-3,-4),(4,3,2,1,-2,-3,-4) 9 38 1, (3,2,-3), (3,2,-3), 2, (2,1,-2),3,(3,2,-3) 9 49 1, (3,2,-3),1,1,2,(2,1,-2), 3 10 49 (2,1,-2),(2,1,-2),(2,1,-2),(2,1,-2),1,(3,2,-3),(3,2,-3),2,3 10 53 1,2,(3,2,1,-2,-3),(2,1,-2),(4,3,-4),(4,3,-4),3,4 10 55 1,1,(3,2,-3),(2,1,-2),(4,3,-4),3,4 10 63 1,1,(4,3,2,1,-2,-3,-4),2,(2,1,-2),(2,1,-2),3,4 10 66 1,1,1,(3,2,1,-2,-3), 2,(2,1,-2),(2,1,-2),3,3 10 80…”

A Bennequin-Type Inequality and Combinatorial Bounds

“…Let L and L 0 be two oriented links in R 3 . Then, the following properties hold (1) if Σ is a weak cobordism between L and L 0 , then |s(L; F) − s(L 0 ; F)| ≤ −χ(Σ); (2) where # denotes the connected sum, m() denotes the mirror image and is the number of components of L. In particular, s is a strong concordance invariant.…”

“…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.…”

“…As an application of Theorem 3, using the computations of the Seifert genus of almost-positive links due to Stoimenov ( [27]), we are able to recover a result due to Tagami ([28]), in the case of knots, and Abe-Tagami ([2]), general case. Corollary 4 ( [2,28]). Let L be an almost-positive non-split link.…”