All Outputs (34)

Embedded surfaces with infinite cyclic knot group (2023)
Journal Article
Conway, A., & Powell, M. (2023). Embedded surfaces with infinite cyclic knot group. Geometry & Topology, 27(2), 739-821. https://doi.org/10.2140/gt.2023.27.739

We study locally flat, compact, oriented surfaces in 4-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus g, to be related by an ambient homeomorphism, a... Read More about Embedded surfaces with infinite cyclic knot group.

Simply-connected manifolds with large homotopy stable classes (2022)
Journal Article
Conway, A., Crowley, D., Powell, M., & Sixt, J. (2023). Simply-connected manifolds with large homotopy stable classes. Journal of the Australian Mathematical Society, 115(2), 172-203. https://doi.org/10.1017/s1446788722000167

For every k ≥ 2 and n ≥ 2 we construct n pairwise homotopically inequivalent simply-connected, closed 4k-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stab... Read More about Simply-connected manifolds with large homotopy stable classes.

The Z-Genus of Boundary Links (2022)
Journal Article
Feller, P., Park, J., & Powell, M. (2023). The Z-Genus of Boundary Links. Revista Matemática Complutense, 36(1), 1-25. https://doi.org/10.1007/s13163-022-00424-3

The Z-genus of a link L in S3 is the minimal genus of a locally at, embedded, connected surface in D4 whose boundary is L and with the fundamental group of the complement innite cyclic. We characterise the Z-genus of boundary links in terms of their... Read More about The Z-Genus of Boundary Links.

Four-manifolds up to connected sum with complex projective planes (2022)
Journal Article
Kaprowski, D., Powell, M., & Teichner, P. (2022). Four-manifolds up to connected sum with complex projective planes. American Journal of Mathematics, 144(1), 75-118. https://doi.org/10.1353/ajm.2022.0001

Based on results of Kreck, we show that closed, connected 4- manifolds up to connected sum with copies of the complex projective plane are classified in terms of the fundamental group, the orientation character and an extension class involving the se... Read More about Four-manifolds up to connected sum with complex projective planes.

Embedding spheres in knot traces (2021)
Journal Article
Feller, P., Miller, A. N., Nagel, M., Orson, P., Powell, M., & Ray, A. (2021). Embedding spheres in knot traces. Compositio Mathematica, 157(10), 2242-2279. https://doi.org/10.1112/s0010437x21007508

The trace of the n-framed surgery on a knot in S3 is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose compleme... Read More about Embedding spheres in knot traces.

Characterisation of homotopy ribbon discs (2021)
Journal Article
Conway, A., & Powell, M. (2021). Characterisation of homotopy ribbon discs. Advances in Mathematics, 391, Article 107960. https://doi.org/10.1016/j.aim.2021.107960

Let Γ be either the infinite cyclic group Z or the Baumslag-Solitar group Zn Z[ 1 2 ]. Let K be a slice knot admitting a slice disc D in the 4-ball whose exterior has fundamental group Γ. We classify the Γ-homotopy ribbon slice discs for K up to topo... Read More about Characterisation of homotopy ribbon discs.

Two-solvable and two-bipolar knots with large four-genera (2021)
Journal Article
Cha, J. C., Miller, A. N., & Powell, M. (2021). Two-solvable and two-bipolar knots with large four-genera. Mathematical Research Letters, 28(2), 331-382. https://doi.org/10.4310/mrl.2021.v28.n2.a2

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that 2-solvable knots are in particular algebraically slice and have vanishing Casson–Gordon obstructions. Similarly all known smooth... Read More about Two-solvable and two-bipolar knots with large four-genera.

Doubly slice knots and metabelian obstructions (2021)
Journal Article
Orson, P., & Powell, M. (2022). Doubly slice knots and metabelian obstructions. Journal of Topology and Analysis, 14(4), 847-873. https://doi.org/10.1142/s1793525321500229

An n-dimensional knot Sn⊂Sn+2 is called doubly slice if it occurs as the cross section of some unknotted (n+1)-dimensional knot. For every n it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dim... Read More about Doubly slice knots and metabelian obstructions.

Homotopy ribbon concordance and Alexander polynomials (2020)
Journal Article
Friedl, S., & Powell, M. (2020). Homotopy ribbon concordance and Alexander polynomials. Archiv der Mathematik, 115(6), 717-725. https://doi.org/10.1007/s00013-020-01517-5

We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L, then the Alexander polynomial of L divides the Alexander polynomial of J.

Triple linking numbers and surface systems (2020)
Journal Article
Davis, C. W., Nagel, M., Orson, P., & Powell, M. (2020). Triple linking numbers and surface systems. Indiana University Mathematics Journal, 69(7), 2505-2547. https://doi.org/10.1512/iumj.2020.69.8081

We give a refined value group for the collection of triple linking numbers of links in the 3–sphere. Given two links with the same pairwise linking numbers we show that they have the same refined triple linking number collection if and only if the li... Read More about Triple linking numbers and surface systems.

Stabilization distance between surfaces (2019)
Journal Article
Miller, A. N., & Powell, M. (2019). Stabilization distance between surfaces. L’Enseignement mathématique, 65(3/4), 397-440. https://doi.org/10.4171/lem/65-3/4-4

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative int... Read More about Stabilization distance between surfaces.

A family of freely slice good boundary links (2019)
Journal Article
Cha, J. C., Kim, M. H., & Powell, M. (2020). A family of freely slice good boundary links. Mathematische Annalen, 376(3-4), 1009-1030. https://doi.org/10.1007/s00208-019-01907-3

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two... Read More about A family of freely slice good boundary links.

Whitney towers and abelian invariants of knots (2019)
Journal Article
Cha, J. C., Orr, K., & Powell, M. (2020). Whitney towers and abelian invariants of knots. Mathematische Zeitschrift, 294(1-2), 519-553. https://doi.org/10.1007/s00209-019-02293-x

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for co... Read More about Whitney towers and abelian invariants of knots.

Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance (2018)
Journal Article
Miller, A. N., & Powell, M. (2018). Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance. Algebraic & geometric topology, 18(6), 3425-3476. https://doi.org/10.2140/agt.2018.18.3425

We give a formula for the duality structure of the 3 –manifold obtained by doing zero-framed surgery along a knot in the 3 –sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blan... Read More about Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance.

Satellites and concordance of knots in 3-manifolds (2018)
Journal Article
Friedl, S., Nagel, M., Orson, P., & Powell, M. (2019). Satellites and concordance of knots in 3-manifolds. Transactions of the American Mathematical Society, 371(4), 2279-2306. https://doi.org/10.1090/tran/7313

Given a 3–manifold Y and a free homotopy class in [S1, Y ], we investigate the set of topological concordance classes of knots in Y × [0, 1] representing the given homotopy class. The concordance group of knots in the 3–sphere acts on this set. We sh... Read More about Satellites and concordance of knots in 3-manifolds.

Smooth and topological almost concordance (2018)
Journal Article
Nagel, M., Orson, P., Park, J., & Powell, M. (2019). Smooth and topological almost concordance. International Mathematics Research Notices, 2019(23), 7324-7355. https://doi.org/10.1093/imrn/rnx338

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Y × [0, 1] and the action of the concordance group of knot... Read More about Smooth and topological almost concordance.

Twisted Blanchfield pairings and decompositions of 3-manifolds (2017)
Journal Article
Friedl, S., Leidy, C., Nagel, M., & Powell, M. (2017). Twisted Blanchfield pairings and decompositions of 3-manifolds. Homology, Homotopy and Applications, 19(2), 275-287. https://doi.org/10.4310/hha.2017.v19.n2.a14

We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation ϕ:Z[π1(Y)]→R, infected by a knot J alo... Read More about Twisted Blanchfield pairings and decompositions of 3-manifolds.

Stable classification of 4-manifolds with 3-manifold fundamental groups (2017)
Journal Article
Kasprowski, D., Land, M., Powell, M., & Teichner, P. (2017). Stable classification of 4-manifolds with 3-manifold fundamental groups. Journal of Topology, 10(3), 827-881. https://doi.org/10.1112/topo.12025

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w2-type and their equivariant intersectio... Read More about Stable classification of 4-manifolds with 3-manifold fundamental groups.

Grope metrics on the knot concordance set (2017)
Journal Article
Cochran, T. D., Harvey, S., & Powell, M. (2017). Grope metrics on the knot concordance set. Journal of Topology, 10(3), 669-699. https://doi.org/10.1112/topo.12018

To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudo-metrics d q , refining the slice genus metric, on the set of concor... Read More about Grope metrics on the knot concordance set.

Concordance Invariance of Levine-Tristram Signatures of Links (2017)
Journal Article
Nagel, M., & Powell, M. (2017). Concordance Invariance of Levine-Tristram Signatures of Links. Documenta Mathematica, 22, 25-43

We determine for which complex numbers on the unit circle the Levine-Tristram signature and the nullity give rise to link concordance invariants.