Papers

A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.

Keller's strong form of the homological conjectures asserts: any finite dimensional algebra cogenerates its unbounded derived category of modules. Here we record an example of a (coconnective) dg algebra with finite dimensional cohomology, which does not cogenerate its module category, along with some related phenomena: a smooth dg category whose dualizing bimodule fails to be nondegenerate, and a nontrivial fully faithful left Calabi-Yau morphism. All examples and most proofs were produced by ChatGPT. In an appendix we explain the reason we were looking for such examples: their existence would follow from the existence of Weinstein symplectic manifolds which failed to satisfy Arnol'd's chord conjecture.

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product $S^2\times S^2$ with two fixed points. The argument starts with a four-fixed-point map, passes to rescaled cotangent coordinates near the antidiagonal, and reduces the local fixed-point equations to two planar saddle problems. A relative modification merges each pair of saddles, and mixed generating functions are used to transfer the modification to the original map. We include detailed explanations of the normal form along a level arc, the Moser equation, uniform existence of mixed generating functions, the order of parameter choices, and the global graph argument. This construction gives a counterexample to the degenerate version of Arnold's conjecture proposed by Arnold\cite{Arnold86} when the manifold is standard $S^2\times S^2$. This work is done with ChatGPT. The author gave the idea to study the limit behavior of $\Phi_\rho$. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.

Chu proved that a symplectic structure $\omega$ on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by $\omega(x\scriptstyle{\Delta}_{\omega} y,z)=-\omega(y,[x,z])$. El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra $\mathfrak{g}$ induces the left-symmetric structure, which is defined by using a linear isomorphism from $\mathfrak{g}$ to $\mathfrak{g}^*$ associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form $\phi$ on a Lie algebra $\mathfrak{g}$, we give a necessary and sufficient condition for the product $\scriptstyle{\Delta}_{\phi}$ on $\mathfrak{g}$ defined by $\phi(x\scriptstyle{\Delta}_{\phi} y,z)=-\phi(y,[x,z])$ to be a left-symmetric structure. We also prove that the left-symmetric structure $\scriptstyle{\Delta}_{\phi}$ is complete if and only if the Lie algebra $\mathfrak{g}$ is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of $\#_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\leq 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer–Vietoris presentation, and a regular cyclic cover of degree $\operatorname{lcm}(p,q)$. The boundary subgroup $\pi_1(\Sigma_G)\times\mathbb Z$ embeds in the fundamental group; this gives a nontrivial Bass–Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

Motivated by recent developments in generalized Kähler geometry, we study morphisms of double Lie groupoids from a simplicial viewpoint. We show that the codiagonal functor $\Wbar$ extends from objects to horizontal and vertical principal bibundles and to square-shaped $(1,1)$-morphisms. The latter determine 2-dimensional morphisms of square and globular shape between anafunctors of Lie 2-groupoids. We illustrate these constructions through two applications: First, we revisit the equivalence between two models of nonabelian gerbes: groupoid bundle gerbes and principal 2-bundles. We show that a principal 2-bundle and its associated groupoid bundle gerbe determine equivalent anafunctors from a manifold to the corresponding structure Lie 2-group. Second, we refine the integration of Manin triples in transitive Courant algebroids. We show that integrations associated with different choices of suitably transverse Manin triples in the same Courant algebroid are related by symplectic Morita equivalences. Consequently, the resulting integration of the background Courant algebroid is well defined up to symplectic Morita equivalence.

We study corank-two distributions $D$ of odd rank $2N+1$ whose associated pencil of skew-symmetric forms $\{d\alpha|_{D}(q)\mid \alpha\in\Omega^1(M),\ \alpha|_{D}=0\}$ lies, at every point, in the generic orbit of the natural $\mathrm{GL}\bigl(D(q)\bigr)$-action; equivalently, $D$ has maximal first Kronecker index. In corank two, this class is the analogue of contact and even-contact distributions. We prove that the corresponding differential relation is ample and hence satisfies the multi-parametric $C^0$-close $h$-principle, and we deduce a complete topological characterization for a $(2N+3)$-dimensional manifold to admit such a distribution. The criterion is satisfied by non-parallelizable manifolds as well. To the best of our knowledge, ampleness has been verified so far for two classes of distributions defined by an open $\mathrm{Diff}$-invariant relation: even-contact distributions (McDuff, 1987) and distributions of rank greater than two with maximal small growth vector, in arbitrary ambient dimension (Martínez-Aguinaga, 2026); for $N\ge2$, our class is a proper open subclass of the latter, singled out by a further condition on the pencil. The verification of ampleness reduces, via Kronecker–Weierstrass normal forms of the restrictions of the pencil to hyperplanes, to computing the convex hulls of the connected components of the complement of the resultant hypersurface in the space of pairs of real binary forms of fixed degrees, components separated by a winding number, i.e., by the classical Cauchy index. This is in contrast with the even-contact case of McDuff, where the set to be removed is thin (i.e., of codimension at least two), so that its complement is automatically connected and ampleness is immediate.

We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes \chi and b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree $\le 4$, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.

We import the theory of $R$-links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of Lagrangian and symplectic sliceness, and more generally establishes one approach to what we call the Lagrangian slice-ribbon conjecture. Our main theorem asserts that a Legendrian knot with Thurston-Bennequin invariant $-1$ is Lagrangrian slice (resp. regularly Lagrangian slice) if and only if it supports a tight transverse (resp. Legendrian) $R$-link derivative.

For a compact connected Hamiltonian $T$-space (with $T$ a torus), the components of the set of (relative) regular values of the momentum map are open convex polytopes, whose closures (called the chambers) form a polyhedral decomposition of the momentum polytope. In a previous paper we showed that there is a canonical affine stratification of the momentum polytope with the property that, while varying through a stratum, the fibers of the momentum map do not change as $T$-spaces. In this paper we show that the strata of this stratification are the relative interiors of the faces of the chambers. In doing so, we also give a proof of the fact that these faces form a polyhedral complex, which (it seems) was missing from the literature. We further extend the above to Hamiltonian $T$-spaces whose momentum map is proper as map into a convex set and, in the compact Kähler case, we point out a description of the above strata in terms of the $T_\mathbb{C}$-orbit closure polytopes.

For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold. When the moduli space is nonempty, there is a natural homomorphism from the mapping class group of the punctured surface to the symplectic mapping class group of this moduli space. It is shown that this homomorphism is injective if and only if the dimension of the moduli space is greater than $2$. This generalizes work of Seidel and Wehrheim–Woodward. Also given is a complete classification of Lagrangian spheres in the projective plane blown up at $5$ points with its monotone symplectic structure, which is the moduli space for the 5-punctured sphere. Furthermore, it is determined when two such Lagrangian spheres can be displaced by a symplectic isotopy. Results are also obtained regarding Lagrangian spheres in the intersection of two quadrics in $\mathbb{C}\mathbb{P}^5$. The proofs involve instanton Floer theory and results on Heegaard splittings. A main technical result establishes the approximation of any Hamiltonian isotopy of the $SU(2)$ moduli space by holonomy perturbations which are used in instanton homology.

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

We study lattice-counting functions on compact integral-integral affine manifolds with corners, which generalize integral unimodular polytopes. We prove analogues of Ehrhart's theorem and Ehrhart-Macdonald reciprocity in this setting.

Welschinger showed that counts of connected holomorphic disks with Lagrangian boundary in symplectic 6-manifolds, meeting at least one boundary constraint, can be made invariant by correcting them with counts of disconnected disks weighted by certain "self-linking" numbers. We show his invariant is the lowest order term in an all-genus curve count where curves are weighted by the Conway polynomials of their boundaries. This in turn is a specialization of the skein-valued curve count, but can be defined without the 4-chain and vector field used in that setup.

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

We describe a polytopal realization of categorical $n$-associahedra. The normal fan of this polytopal realization is a modification of the authors' velocity fan and was found by OpenAI's Astra model.

Let $F$ be a singular fibre of a relatively minimal complex elliptic fibration with smooth total space, and let $\Omega$ be a nonvanishing holomorphic two-form near $F$. We show that a small neighbourhood of $F$ is a Weinstein domain for $\mathrm{Re}\,\Omega$ whose completion is a Legendrian surgery, with cocores obtained by completing holomorphic disks transverse to the components of $F$. For every coefficient field and every multiplicative bulk class, we compute the wrapped Floer cohomology of these cocores and prove that it is concentrated in degree zero. The cocores generate, so the bulk-deformed wrapped Fukaya category is equivalent to the category of perfect modules over an explicit algebra: a multiplicative preprojective algebra of affine type for the normal crossing fibres, and a quiver algebra with relations for types $II$, $III$ and $IV$. Applications include formality of the affine plumbing dg algebras; mirrors given by resolved affine surfaces at every classical bulk class for affine D, E and types II, III, IV; and mirrors given by quotient stacks of algebraic tori at root-of-unity bulk classes for the four affine stars.

In this paper, we investigate higher-order symmetries and non-geometric symmetries within the framework of $k$-cosymplectic Hamiltonian field theory. We introduce the concept of generalized infinitesimal symmetry, which relaxes the standard symmetry condition, and then define two new classes of symmetries: higher-order general infinitesimal Cartan symmetries and good non-geometric symmetries. For each of these symmetry classes, we establish a Noether-type theorem that provides explicit conserved quantities for exact $k$-cosymplectic Hamiltonian systems. These results extend and generalize previous work on standard Cartan symmetries in the $k$-cosymplectic setting. Furthermore, we extend all of these results in a parallel fashion to Hamiltonian systems on $k$-symplectic manifolds, demonstrating the unifying nature of our approach. Concrete examples are provided to illustrate the existence and applicability of the proposed symmetries. Our framework offers a systematic geometric treatment of generalized symmetries and their associated conservation laws in classical field theories.

Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $\beta$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $\beta$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,\beta)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $\beta$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais–Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.

We develop a root-theoretic localization formalism for genus-zero Gromov–Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov–Kontsevich–Pantev–Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in $\operatorname{LG}(3,6)$; the very general smooth quadric section in $G_2/P_2$; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.

This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross–Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi–Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev–Borisov toric degenerations, as well as some toric degenerations of Calabi–Yau threefolds coming from the Gross–Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi–Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{\pi_A} d'(A;K). \end{align*} Here $\prec_{\pi_A}$ represents block-majorization corresponding to the partition $\pi_A$ of $\{1,\ldots, n\}$ given by the equality blocks of symplectic eigenvalues of $A$. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.

We introduce a matrix generalization of the cone of Harvey and Lawson and prove that it is an exact special Lagrangian manifold. We further show that it belongs to a family of exact special Lagrangian manifolds that foliate the balanced manifold arising in deep learning.

Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y^m=x^n$ in $\mathbb{C}^2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.

Let $V$ be a complex symplectic vector space and let $\mathbb{P}(V)$ carry the induced contact structure and symplectic polarity. We study the loci of hypersurfaces containing codimension-two cones supported on hyperplanes. The starting point is the following characterization: a projective subvariety $X\subset\mathbb{P}(V)$ which is a hypersurface of degree at least two in a hyperplane $H$ is involutive if and only if it is a cone whose vertex contains the polar point $\sigma(H)$. We give a short proof, construct the corresponding incidence spaces, compute their dimensions and express the incidence degrees as Segre-class integrals. In $\mathbb{P}^3$ the incidence map is birational for every $d\ge m\ge 2$ except $(m,d)=(2,2)$, and the same holds in $\mathbb{P}^5$; the quadratic case is exceptional in every dimension, the incidence having generic degree $2n$ in $\mathbb{P}^{2n-1}$. We also give a sufficient criterion for birationality in higher dimension, closed formulas in $\mathbb{P}^3$ and coefficient formulas in $\mathbb{P}^5$. For quadric cones in $\mathbb{P}^5$ the degree is the product of shifted binomial factors and an irreducible polynomial of degree 36, and we explain where the shifted factors come from.

The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$ and $\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $\pi_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $\pi_1$.

In this paper, we study de Rham realizations for finite orientifold quotients. On the orientation preserving inertia of an almost complex global quotient we construct an anti-linear involution of the Chen-Hu algebra whose fixed subalgebra is a real form of the Chen-Ruan algebra. For genuine orientation reversing fixed sectors we prove that no coordinate invariant single sector fractional age exists. When a Calabi-Yau volume line is preserved, the relative phases define a canonical class $[\mu_{\mathbb R}]\in H^2(\widehat G;\mathbb Z_\epsilon)$. We identify this class with the Bockstein obstruction to a coherent real phase lift and prove that its vanishing is equivalent, after rephasing the volume form, to a phase trivial orientifold action. We then formulate a general Euler-Gysin realization theorem using relative Fredholm bundles and determinant-line sewing. For phase trivial Calabi-Yau surfaces we verify the admissibility conditions directly and obtain a canonically defined associative graded cohomology algebra containing the Chen-Hu even sector and the odd fixed surfaces. For an Eisenstein abelian surface with an order six dihedral action we compute the centralizer invariant algebra explicitly; it is a $16$ dimensional Frobenius algebra whose odd sector multiplication is noncommutative in the shifted grading. We also explain the relation with earlier unoriented orbifold cohomology constructions.

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

We compute the geometric monodromy factorization of Xiao's genus-two Lefschetz fibration directly from its branched-cover construction, following Moishezon's braid-monodromy method. The complete quadrangle determines the motion of the six branch points and gives four nonseparating and three separating vanishing cycles. At each of the three $3+3$ degenerations, the branch motion determines the spherical mapping class, and Xiao's local holomorphic model determines its genus-two lift. After fixing a distinguished system of paths, we obtain an ordered positive factorization of type $(4,3)$ and give Artin coordinates for the seven factors.