The defect of Bennequin-Eliashberg inequality and Bennequin surfaces

Abstract

For a null-homologous transverse link T in a general contact manifold with an open book, we explore strongly quasipositive braids and Bennequin surfaces. We define the defect δ(T ) of the Bennequin-Eliashberg inequality.We study relations between δ(T ) and minimal genus Bennequin surfaces of T . In particular, in the disk open book case, under some large fractional Dehn twist coefficient assumption, we show that δ(T ) = N if and only if T is the boundary of a Bennequin surface with exactly N negatively twisted bands. That is, the Bennequin inequality is sharp if and only if it is the closure of a strongly quasipositive braid.

Positivities of Knots and Links and the Defect of Bennequin Inequality

“…Proof. Corollary 1.10 in [23] implies that there exists a 3-braid representative K of K in the band generators {σ i,j | 1 ≤ i < j ≤ 3} such that the associated Bennequin surface Σ K realizes the maximal Euler characteristic χ(K) and contains δ(K) = 1 negative band. Using [37, Theorem 6] of Xu, we may assume that K is represented by a braid word having the form N P ; namely,…”

“…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…”

“…The following conjecture on links with δ 3 (K) = 0 can be found in [23] as "Stronger Form of Conjecture 2". Baader [3] • The Bennequin inequality (1.2) is sharp for K;…”

Quasi-right-veering braids and nonloose links

“…Here we present an algebraic formulation so that the connection to distinguished monodromy is clear. For a geometric formulation based on open book foliation machinery, we refer the paper [24].…”