Publications (25)
The relative Whitney trick and its applications
Christopher William Davis, Patrick Orson, JungHwan Park
We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a -manifold with boundary.…
Smooth and topological almost concordance
Matthias Nagel, Patrick Orson, JungHwan Park +1
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 con…
Satellites and concordance of knots in 3-manifolds
Stefan Friedl, Matthias Nagel, Patrick Orson +1
Given a 3-manifold and a free homotopy class in , we investigate the set of topological concordance classes of knots in representing the given homotop…
Smoothing topological pseudo-isotopies of 4-manifolds
Patrick Orson, Mark Powell, Oscar Randal-Williams
Given a closed, smooth 4-manifold and self-diffeomorphism that is topologically pseudo-isotopic to the identity, we study the question of whether is moreover smoothly p…
Locally flat simple spheres in
Anthony Conway, Patrick Orson
The fundamental group of the complement of a locally flat surface in a -manifold is called the knot group of the surface. In this article we prove that two locally flat -sphe…
Unknotting nonorientable surfaces
Anthony Conway, Patrick Orson, Mark Powell
Given a nonorientable, locally flatly embedded surface in the -sphere of nonorientable genus , Massey showed that the normal Euler number lies in $\lbrace -2h,-2h+4,\ldots,2h…
Abelian invariants of doubly slice links
Anthony Conway, Patrick Orson
We provide obstructions to a link in arising as the cross section of any number of unlinked spheres in . Our obstructions arise from the multivariable signature, the Bla…
Simple spines of homotopy 2-spheres are unique
Patrick Orson, Mark Powell
A locally flatly embedded -sphere in a compact -manifold is called a spine if the inclusion map is a homotopy equivalence. A spine is called simple if the complement of t…
Double -groups and doubly-slice knots
Patrick Orson
We develop a theory of chain complex double-cobordism for chain complexes equipped with Poincaré duality. The resulting double-cobordism groups are a refinement of Ranicki's torsi…
Double Witt groups
Patrick Orson
The difference between slice and doubly-slice knots is reflected in algebra by the difference between metabolic and hyperbolic Blanchfield linking forms. We exploit this algebraic…
Khovanov homotopy calculations using flow category calculus
Andrew Lobb, Patrick Orson, Dirk Schuetz
The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first t…
A calculus for flow categories
Andrew Lobb, Patrick Orson, Dirk Schuetz
We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow cate…
A Khovanov stable homotopy type for colored links
Andrew Lobb, Patrick Orson, Dirk Schuetz
We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a colo…
Null, recursively starlike-equivalent decompositions shrink
Jeffrey Meier, Patrick Orson, Arunima Ray
A subset of a metric space is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into for some , sending to a s…
A lower bound for the doubly slice genus from signatures
Patrick Orson, Mark Powell
The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the c…
Double L-theory
Patrick Orson
We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal…
Triple linking numbers and surface systems
Christopher William Davis, Matthias Nagel, Patrick Orson +1
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 t…
Embedding spheres in knot traces
Peter Feller, Allison N. Miller, Matthias Nagel +3
The trace of -framed surgery on a knot in is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a mani…
Framed cobordism and flow category moves
Andrew Lobb, Patrick Orson, Dirk Schuetz
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex w…
Simply slicing knots
Anthony Conway, Patrick Orson, Mark Pencovitch
Given a simply-connected 4-manifold with boundary the 3-sphere, this paper establishes sufficient conditions for a knot in the boundary to be sliced by a locally flat disc in the 4…
Doubly slice knots and metabelian obstructions
Patrick Orson, Mark Powell
For , we develop -signature obstructions for -dimensional knots with metabelian knot groups to be doubly slice. For each , we construct an infi…
Smooth stable isotopy of topologically isotopic surfaces
Daniel Galvin, Patrick Orson, Mark Powell
A stabilisation of a -manifold is the connected sum of with some number of copies of . If two smooth surfaces in a -manifold are topologically isotopic…
Twist spinning of knots and metabolizers of Blanchfield pairings
Stefan Friedl, Patrick Orson
In a classic paper Zeeman introduced the k-twist spin of a knot K and showed that the exterior of a twist spin fibers over S^1. In particular this result shows that the knot K # -K…
A survey of the foundations of four-manifold theory in the topological category
Stefan Friedl, Matthias Nagel, Patrick Orson +1
This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise r…
Mapping class groups of simply connected 4-manifolds with boundary
Patrick Orson, Mark Powell
We compute the topological mapping class group of every compact, simply connected, topological 4-manifold. This was previously only known in the closed case. If the 4-manifold is s…