Lefschetz fibration structures on knot surgery 4-manifolds

Abstract

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots K. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

Charts, signatures, and stabilizations of Lefschetz fibrations

“…For example, the Lefschetz fibration on E(n) K constructed by Fintushel and Stern [9, Theorem 14] (see also Park and Yun [37]) for a fibered knot K becomes isomorphic to that on E(n) K ′ for another fibered knot K ′ of the same genus after one stabilization. Similar results hold for Lefschetz fibrations on Y(n; K 1 , K 2 ) constructed by Fintushel and Stern [9, §7] (see also Park and Yun [38]) as well as fiber sums of (generalizations of) Matsumoto's fibration studied by Ozbagci and Stipsicz [36], Korkmaz [24,25], and Okamori [35].…”

“…Nosaka [34] has recently defined an invariant which is not additive under fiber sum. Non-numerical invariants such as monodromy group would be also useful (see Matsumoto [32] and Park and Yun [37,38]).…”

Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration

“…We call this Lefschetz fibration the Matsumoto-Cadavid-Korkmaz Lefschetz fibration (MCK for short) in this paper. The MCK Lefschetz fibration has become one of the most basic examples in the theory of Lefschetz fibrations and played great roles, especially as a powerful source to construct new Lefschetz fibrations, surface bundles, Stein fillings, symplectic 4-manifolds, and so on, with various interesting features [32,23,24,38,33,17,40,35,36,4,1,2,3,5,9,20,21,22,29]. The MCK Lefschetz fibration itself has several remarkable features such as having quite small number of critical points (the smallest among the known examples for g ≥ 4), large b 1 (the largest among the known for even g 1 ), high symmetricity of the vanishing cycles, in particular, it 1 For odd g it had been also the largest until Baykur [6] recently found a Lefschetz fibration with b 1 one larger than that of the MCK Lefschetz fibration.…”