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arXiv 
Instantons, special cycles, and knot concordance (with Ali Daemi, Hayato Imori, Kouki Sato, Masaki Taniguchi)


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arxiv

Abstract: We introduce a framework for defining concordance invariants of knots using equivariant singular instanton Floer theory with ChernSimons 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 twobridge knots, and more. In particular, we prove a quasiadditivity property of s^{#}, answering a question of Gong. We also introduce invariants that are formally similar to the Heegaard Floer τinvariant of Oszváth and Szabó and the εinvariant 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 PinzónCaicedo. Second, we produce infinitely many twobridge knots K which are torsion in the algebraic concordance group and yet have the property that the set of positive 1/nsurgeries on K is a linearly independent set in the homology cobordism group. Finally, for a knot which is quasipositive 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.

ChernSimons functional, singular instantons, and the fourdimensional clasp number (with Ali Daemi)


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arxiv

Abstract: Kronheimer and Mrowka asked whether the difference between the fourdimensional 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 ChernSimons 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 quasialternating 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 nonzero signature, it is shown that there is a traceless representation of the concordance complement which restricts to nontrivial representations of the knot groups. Finally, some evidence towards an extension of the sliceribbon conjecture to torus knots is provided.

Associative submanifolds and gradient cycles (with Simon Donaldson)


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arxiv

Abstract: We discuss a model for associative submanifolds in G2manifolds with K3 fibrations, in the adiabatic limit. The model involves graphs in a 3manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds.

Framed instanton homology of surgeries on Lspace knots (with Tye Lidman, Juanita PinzonCaicedo)


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arxiv

Abstract: For knots with an instanton Lspace 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 Lspace surgery, then the theories agree for all integral surgeries. In order to prove the main result, we compute the mod 2 grading of the BaldwinSivek contact invariant in framed instanton homology.

Equivariant aspects of singular instanton Floer homology (with Aliakbar Daemi)


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arxiv

Abstract: We associate several invariants to a knot in an integer homology 3sphere 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 ChernSimons 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 Froyshovtype invariants which are concordance invariants, and more.

Computing nuinvariants of Joyce's compact G2manifolds


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arxiv

Abstract: Crowley and Nordström introduced an invariant of G2structures on the tangent bundle of a closed 7manifold, 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 torsionfree G2manifolds defined by Joyce using a generalized Kummer construction.

Niemeier lattices, smooth 4manifolds and instantons arXiv:1808.10321


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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. Trans. Amer. Math. Soc. 372 (2019), no. 11, 78057829. (with Marco Golla)

Trans. Amer. Math. Soc.

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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


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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. Adv. Math. 367 (2020). (with Sucharit Sarkar and Matt Stoffregen)

Adv. Math.

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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.

Newstead's MayerVietoris argument in characteristic 2. Internat. J. Math. 30 (2019), no. 12, 1950065, 18 pp. (with Matt Stoffregen)

Internat. J. Math.

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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. Canad. J. Math. 71 (2016), no. 3, 683715. (with Matt Stoffregen)

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.
