Lefschetz fibrations on knot surgery 4-manifolds via Stallings twist (Q2408338)

If \(X\) is a simply connected smooth 4-manifold containing an embedded torus \(T\) of square \(0\) and \(\pi_1(X\smallsetminus T)=1\), then \(X_K=X\#_{T=T_m}S^1\times M_K=[X\smallsetminus(T\times D^2)]\cup[S^1\times(S^3\smallsetminus N(K))]\) is said to be Fintushel-Stern's knot surgery 4-manifold for any knot \(K\subset S^3\), where \(M_K\) is the 3-manifold obtained by performing a 0-framed surgery along \(K\). \(X_4\) is constructed by taking a fiber sum along a torus \(T\) in \(X\) and \(T_m=S^1\times m\) in \(S^1\times M_K\), with the requirement that in the second expression the two pieces are glued together in such a way that the homology class \([pt\times\partial D^2]\) is identified with \([pt\times l]\), and \(m\) and \(l\) are the meridian and longitude of \(K\), respectively. If \(X\) is a compact, oriented smooth 4-manifold, then a Lefschetz fibration is a proper smooth map \(\pi:X\to B\), where \(B\) is a compact connected oriented surface and \(\pi^{-1}(\partial B)=\partial X\) such that the set of critical points \(C=\{p_1,p_2,\dots,p_n\}\) of \(\pi\) is nonempty and lies in \(\text{int}(X)\), and for each \(p_i\) and \(b_i=\pi(p_i)\), there are local complex coordinate charts agreeing with the orientations of \(X\) and \(B\) such that \(n\) can be expressed as \(\pi(z_1,z_2)=z_1^2+z_2^2\). The knot surgery 4-manifold \(X_K\) is homeomorphic, but not diffeomorphic, to a given \(X\), and if \(X\) is a simply connected elliptic surface \(E(2)_K\), \(T\) is a generic elliptic fiber, and \(K\) is a fibered knot in \(S^3\), then the knot surgery 4-manifold \(E(2)_K\) admits not only a symplectic structure, but also a genus \(2g(K)+1\) Lefschetz fibration structure. In this paper, the authors show that for each integer \(n>0\) some instances of \(E(2)_K\) admit \(2^n\) nonisomorphic Lefschetz fibration structures. To present such examples, they first perform a knot surgery on \(E(2)\) using a connected sum of \(n\) copies of genus 2 fibered knots, which are obtained by Stallings twists from the square knot \(3_1\#3_1^*\), then they consider the corresponding monodromy factorizations and monodromy groups. The main result of the paper states that for each integer \(n>0\) and \((m_1,m_2,\dots,m_n)\in\mathbb Z^n\), a knot surgery 4-manifold admits \(2^n\) nonisomorphic genus \((4n+1)\) Lefschetz fibrations over \(S^2\), where \(K_{m_i}\) denotes a fibered knot obtained by performing \(|m_i|\) left/right handed full twists on the square knot.