Outputs (33)

Splitting Numbers of Links (2017)
Journal Article
Cha, J. C., Friedl, S., & Powell, M. (2017). Splitting Numbers of Links. Proceedings of the Edinburgh Mathematical Society, 60(03), 587-614. https://doi.org/10.1017/s0013091516000420

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and A... Read More about Splitting Numbers of Links.

A Calculation of Blanchfield Pairings of 3-Manifolds and Knots (2017)
Journal Article
Friedl, S., & Powell, M. (2017). A Calculation of Blanchfield Pairings of 3-Manifolds and Knots. Moscow mathematical journal, 17(1), 59-77

We calculate Blanchfield pairings of 3-manifolds. In particular, we give a formula for the Blanchfield pairing of a fibred 3-manifold and we give a new proof that the Blanchfield pairing of a knot can be expressed in terms of a Seifert matrix.

The four-genus of a link, Levine–Tristram signatures and satellites (2016)
Journal Article
Powell, M. (2017). The four-genus of a link, Levine–Tristram signatures and satellites. Journal of Knot Theory and Its Ramifications, 26(02), Article 1740008. https://doi.org/10.1142/s0218216517400089

We give a new proof that the Levine–Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal... Read More about The four-genus of a link, Levine–Tristram signatures and satellites.

Blanchfield forms and Gordian distance (2016)
Journal Article
Borodzik, M., Friedl, S., & Powell, M. (2016). Blanchfield forms and Gordian distance. Journal of the Mathematical Society of Japan, 68(3), 1047-1080. https://doi.org/10.2969/jmsj/06831047

Given a link in S3S3 we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise... Read More about Blanchfield forms and Gordian distance.

Casson towers and slice links (2015)
Journal Article
Cha, J. C., & Powell, M. (2016). Casson towers and slice links. Inventiones Mathematicae, 205(2), 413-457. https://doi.org/10.1007/s00222-015-0639-z

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group... Read More about Casson towers and slice links.

A specious unlinking strategy (2014)
Journal Article
Friedl, S., Nagel, M., & Powell, M. (2014). A specious unlinking strategy. Periodica Mathematica Hungarica, 69(2), 207-210. https://doi.org/10.1007/s10998-014-0063-y

We show that the following unlinking strategy does not always yield an optimal sequence of crossing changes: first split the link with the minimal number of crossing changes, and then unknot the resulting components.

Nonconcordant links with homology cobordant zero-framed surgery manifolds (2014)
Journal Article
Cha, J. C., & Powell, M. (2014). Nonconcordant links with homology cobordant zero-framed surgery manifolds. Pacific Journal of Mathematics, 272(1), 1-33. https://doi.org/10.2140/pjm.2014.272.1

We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topolog... Read More about Nonconcordant links with homology cobordant zero-framed surgery manifolds.

Shrinking of toroidal decomposition spaces (2014)
Journal Article
Kasprowski, D., & Powell, M. (2014). Shrinking of toroidal decomposition spaces. Fundamenta Mathematicae, 227(3), 271-296. https://doi.org/10.4064/fm227-3-3

Given a sequence of oriented links L^1,L^2,L^3,... each of which has a distinguished, unknotted component, there is a decomposition of the 3-sphere naturally associated to it, which is constructed as the components of the intersection of an infinite... Read More about Shrinking of toroidal decomposition spaces.

Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds (2014)
Journal Article
Borodzik, M., & Powell, M. (2016). Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds. Journal of Geometric Analysis, 26(1), 57-87. https://doi.org/10.1007/s12220-014-9538-6

We prove that an embedded cobordism between manifolds with boundary can be split into a sequence of right product and left product cobordisms, if the codimension of the embedding is at least two. This is a topological counterpart of the algebraic spl... Read More about Embedded Morse Theory and Relative Splitting of Cobordisms of Manifolds.

Covering link calculus and the bipolar filtration of topologically slice links (2014)
Journal Article
Cha, J. C., & Powell, M. (2014). Covering link calculus and the bipolar filtration of topologically slice links. Geometry & Topology, 18(3), 1539-1579. https://doi.org/10.2140/gt.2014.18.1539

The bipolar filtration introduced by T Cochran, S Harvey and P Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1–bipolar knots which are not 2–bipolar. For... Read More about Covering link calculus and the bipolar filtration of topologically slice links.