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arXiv 
Niemeier lattices, smooth 4manifolds and instantons arXiv:1808.10321 submitted


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arxiv

Abstract: We show that the set of even positive definite lattices that arise from smooth, simplyconnected 4manifolds bounded by a fixed homology 3sphere can depend on more than the ranks of the lattices. We provide two homology 3spheres with distinct sets of such lattices, each containing a distinct nonempty subset of the rank 24 Niemeier lattices.

On definite lattices bounded by integer surgeries along knots with slice genus at most 2 (with Marco Golla) arXiv:1807.11931 submitted


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arxiv

Abstract: We classify the positive definite intersection forms that arise from smooth 4manifolds with torsionfree homology bounded by positive integer surgeries on the righthanded 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 YangMills instanton gauge theory and Heegaard Floer correction terms.

On definite lattices bounded by a homology 3sphere and YangMills instanton Floer theory arXiv:1805.07875 submitted


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arxiv

Abstract: Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4manifolds bounded by certain homology 3spheres. For example, we show that for +1 surgery on the (2,5) torus knot, the only nondiagonal lattices that can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands. We require that our 4manifolds have no 2torsion in their homology.

An odd Khovanov homotopy type (with Sucharit Sarkar and Matt Stoffregen) arXiv:1801.06308 submitted


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arxiv

Abstract: For each link and every quantum grading, we construct a stable homotopy type whose cohomology recovers OzsvathRasmussenSzabo's odd Khovanov homology, following a construction of LawsonLipshitzSarkar 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.

On Newstead's MayerVietoris argument in characteristic 2 (with Matt Stoffregen) arXiv:1707.06268 submitted


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arxiv

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 MayerVietoris 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.

The cohomology of rank two stable bundle moduli: mod two nilpotency & skew Schur polynomials (with Matt Stoffregen) to appear in Canad. J. of Math.

Canad. J. Math.

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arxiv

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.

Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle. Adv. Math. 336 (2018), 377408. (with Bill Chen)

Adv. Math.

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arxiv

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^264. We then compute the framed instanton homology of a surface times a circle with nontrivial bundle, which is closely related to the kernel of u^264. 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.

Twofold quasialternating links, Khovanov homology and instanton homology. Quantum Topol. 9 (2018), no. 1, 167205. (with Matt Stoffregen)

Quantum Topol.

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arxiv

Abstract: We introduce a class of links strictly containing quasialternating 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 nontrivial real 3plane 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.

Kleinfour connections and the Casson invariant for nontrivial
admissible U(2) bundles. Algebr. Geom. Topol. 17 (2017), no. 5, 28412861. (with Matt Stoffregen)

Algebr. Geom. Topol.

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arxiv

Abstract: Given a rank 2 hermitian bundle over a 3manifold that is nontrivial 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 2divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the 3manifold. This formula counts flat connections on the induced adjoint bundle with Kleinfour holonomy.

Instantons and odd Khovanov homology. J. Topol. 8 (2015), no. 3, 744810. See errata for minor corrections.

J. Topol.

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arxiv

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 3manifold connectsummed with a 3torus. 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.
