Here is a list of my publications and preprints. Click on a title to view the abstract, or
to show all abstracts. Articles are listed in the order in which they were first written, from newest to oldest.
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Unoriented skein exact triangles in equivariant singular instanton Floer theory (with Ali Daemi)
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submitted
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arxiv
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Abstract: Equivariant singular instanton Floer theory is a framework that associates to a knot in an integer homology 3-sphere a suite of homological invariants that are derived from circle-equivariant Morse-Floer theory of a Chern-Simons functional for framed singular SU(2)-connections. These invariants generalize the instanton knot homology of Kronheimer and Mrowka. In the present work, these constructions are extended from knots to links with non-zero determinant, and several unoriented skein exact triangles are proved in this setting. As a particular case, a categorification of the behavior of the Murasugi signature for links under unoriented skein relations is established. In addition to the exact triangles, Froyshov-type invariants for links are defined, and several computations using the exact triangles are carried out. The computations suggest a relationship between Heegaard Floer L-space knots and those knots whose instanton-theoretic categorification of the knot signature is supported in even gradings.
A main technical contribution of this work is the construction of maps for certain cobordisms between links on which obstructed reducible singular instantons are present. These constructions are inspired by recent work of the first author and Miller Eismeier in the setting of non-singular instanton theory for rational homology 3-spheres.
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Rank three instantons, representations and sutures (with Ali Daemi, Nobuo Iida)
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submitted
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arxiv
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Abstract: We show that the knot group of any knot in any integer homology sphere admits a non-abelian representation into SU(3) such that meridians are mapped to matrices whose eigenvalues are the three distinct third roots of unity. This answers the N=3 case of a question posed by Xie and the first author. We also characterize when a PU(3)-bundle admits a flat connection. The key ingredient in the proofs is a study of the ring structure of U(3) instanton Floer homology of a circle times a surface. In an earlier paper, Xie and the first author stated the so-called eigenvalue conjecture about this ring, and in this paper we partially resolve this conjecture. This allows us to establish a surface decomposition theorem for U(3) instanton Floer homology of sutured manifolds, and then obtain the mentioned topological applications. Along the way, we prove a structure theorem for U(3) Donaldson invariants, which is the counterpart of Kronheimer and Mrowka's structure theorem for U(2) Donaldson invariants. We also prove a non-vanishing theorem for the U(3) Donaldson invariants of symplectic manifolds.
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Metric stretching and the period map for smooth 4-manifolds
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submitted
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pdf
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arxiv
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Abstract: The period map for a smooth closed 4-manifold assigns to a Riemannian metric the space of self-dual harmonic 2-forms. This map is from the space of metrics to the Grassmannian of maximal positive subspaces in the second cohomology, where positivity is defined by cup product. We show that the period map has dense image for every 4-manifold, and that it is surjective if b^+=1. Similar results hold for manifolds of dimension a multiple of four. The proofs involve families of metrics constructed by stretching along various hypersurfaces.
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Instantons, special cycles, and knot concordance (with Ali Daemi, Hayato Imori, Kouki Sato, Masaki Taniguchi)
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submitted
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arxiv
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Abstract: We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with Chern-Simons filtration. It is demonstrated that many of the concordance invariants defined using instantons in recent years can be recovered from our framework. This relationship allows us to compute Kronheimer and Mrowka's s#-invariant and fractional ideal invariants for two-bridge knots, and more. In particular, we prove a quasi-additivity property of s#, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer τ-invariant of Oszváth and Szabó and the ε-invariant of Hom. We provide evidence for a precise relationship between these latter two invariants and the s#-invariant.
Some new topological applications that follow from our techniques are as follows. First, we produce a wide class of patterns whose induced satellite
maps on the concordance group have the property that their images have infinite rank, giving a partial answer to a conjecture of Hedden and Pinzón-Caicedo. Second, we produce infinitely many two-bridge knots K which are torsion in the algebraic concordance group and yet have the property that the set of positive 1/n-surgeries on K is a linearly independent set in the homology cobordism group. Finally, for a knot which is quasi-positive and not slice, we prove that any concordance from the knot admits an irreducible SU(2)-representation on the fundamental group of the concordance complement.
While much of the paper focuses on constructions using singular instanton theory with the traceless meridional holonomy condition, we also develop an
analogous framework for concordance invariants in the case of arbitrary holonomy parameters, and some applications are given in this setting.
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Chern-Simons functional, singular instantons, and the four-dimensional clasp number. J. Eur. Math. Soc. (JEMS) 26 (2024), no. 6, 2127--2190. (with Ali Daemi)
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J. Eur. Math. Soc.
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pdf
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arxiv
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Abstract: Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern-Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Froyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless
SU(2) representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.
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Associative submanifolds and gradient cycles. Surveys in differential geometry 2019. Differential geometry, Calabi-Yau theory, and general relativity. Part 2, 39--65, Surv. Differ. Geom., 24, Int. Press, Boston, MA, 2022. (with Simon Donaldson)
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Surveys in Diff. Geom.
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pdf
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arxiv
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Abstract: We discuss a model for associative submanifolds in G2-manifolds with K3 fibrations, in the adiabatic limit. The model involves graphs in a 3-manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds.
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Framed instanton homology of surgeries on L-space knots. Indiana Univ. Math. J. 71 (2022), no. 3, 1317--1347. (with Tye Lidman, Juanita Pinzon-Caicedo)
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Indiana Univ. Math. J.
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arxiv
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Abstract: For knots with an instanton L-space surgery, we compute the framed instanton Floer homology of all integral surgeries. As a consequence, if a knot has a Heegaard Floer and instanton Floer L-space surgery, then the theories agree for all integral surgeries. In order to prove the main result, we compute the mod 2 grading of the Baldwin-Sivek contact invariant in framed instanton homology.
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Equivariant aspects of singular instanton Floer homology (with Aliakbar Daemi)
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To appear in Geom. & Topol.
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pdf
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arxiv
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Abstract: We associate several invariants to a knot in an integer homology 3-sphere using SU(2) singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern-Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Froyshov-type invariants which are concordance invariants, and more.
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Computing nu-invariants of Joyce's compact G2-manifolds
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preprint
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pdf
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arxiv
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Abstract: Crowley and Nordström introduced an invariant of G2-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordström, we compute it for many of the closed torsion-free G2-manifolds defined by Joyce using a generalized Kummer construction.
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Niemeier lattices, smooth 4-manifolds and instantons. Math. Ann. 379 (2021), no. 1-2, 549--568.
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Math. Ann.
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pdf
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arxiv
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Abstract: We show that the set of even positive definite lattices that arise from smooth, simply-connected 4-manifolds bounded by a fixed homology 3-sphere can depend on more than the ranks of the lattices. We provide two homology 3-spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.
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On definite lattices bounded by integer surgeries along knots with slice genus at most 2. Trans. Amer. Math. Soc. 372 (2019), no. 11, 7805--7829. (with Marco Golla)
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Trans. Amer. Math. Soc.
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pdf
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arxiv
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Abstract: We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the (2,5)-torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most two. The proofs use input from Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
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On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory. Geom. Topol. 28 (2024), no. 4, 1587--1628.
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Geom. & Topol.
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pdf
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arxiv
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Abstract: Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot, the only non-diagonal lattices that can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands. We require that our 4-manifolds have no 2-torsion in their homology.
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An odd Khovanov homotopy type. Adv. Math. 367 (2020). (with Sucharit Sarkar and Matt Stoffregen)
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Adv. Math.
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pdf
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arxiv
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Abstract: For each link and every quantum grading, we construct a stable homotopy type whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore, the odd Khovanov homotopy type carries an involution whose fixed point set is a desuspension of the even Khovanov homotopy type. We also construct an involution on an even Khovanov homotopy type, with fixed point set a desuspension of the odd homotopy type.
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Newstead's Mayer-Vietoris argument in characteristic 2. Internat. J. Math. 30 (2019), no. 12, 1950065, 18 pp. (with Matt Stoffregen)
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Internat. J. Math.
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pdf
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arxiv
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Abstract: Consider the moduli space of framed flat U(2) connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any field not of characteristic two. We adapt his method in characteristic two to produce conjectural recursive formulae for the mod two betti numbers of the framed moduli space which we partially verify. We also discuss the interplay with the mod two cohomology ring structure of the unframed moduli space.
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The cohomology of rank two stable bundle moduli: mod two nilpotency & skew Schur polynomials. Canad. J. Math. 71 (2016), no. 3, 683--715. (with Matt Stoffregen)
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Canad. J. Math.
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pdf
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arxiv
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Abstract: We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.
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Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle. Adv. Math. 336 (2018), 377--408. (with Bill Chen)
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Adv. Math.
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pdf
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arxiv
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Abstract: In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism u^2-64. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of u^2-64. We discuss these results in the context of the moduli space of stable rank two holomorphic bundles with fixed odd determinant over a Riemann surface.
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Two-fold quasi-alternating links, Khovanov homology and instanton homology. Quantum Topol. 9 (2018), no. 1, 167--205. (with Matt Stoffregen)
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Quantum Topol.
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pdf
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arxiv
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Abstract: We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted markings on a link with bundle data over the branched cover, we also provide many computations of framed instanton homology in the presence of a non-trivial real 3-plane bundle. We discuss evidence for a spectral sequence from the twisted Khovanov homology of a link with mod 2 coefficients to the framed instanton homology of the double branched cover. We also discuss the relevant mod 4 gradings.
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Klein-four connections and the Casson invariant for non-trivial
admissible U(2) bundles. Algebr. Geom. Topol. 17 (2017), no. 5, 2841--2861. (with Matt Stoffregen)
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Algebr. Geom. Topol.
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pdf
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arxiv
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Abstract: Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the 2-divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the 3-manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy.
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Instantons and odd Khovanov homology. J. Topol. 8 (2015), no. 3, 744--810. See errata for minor corrections.
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J. Topol.
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pdf
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arxiv
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Abstract: We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles.
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William Chen-Mertens (York University), Ali Daemi (Washington University in St. Louis), Simon Donaldson (Imperial College), Marco Golla (Université de Nantes), Nobuo Iida (Tokyo Institute of Technology), Hayato Imori (Kyoto University), Tye Lidman (North Carolina State), Juanita Pinzón-Caicedo (Notre Dame), Sucharit Sarkar (UCLA), Kouki Sato (Meijo University), Matt Stoffregen (Michigan State), Masaki Taniguchi (Kyoto University)
