53 citations · 228 across the 24 of their papers we have counts for
27 papers
Jordan curves inscribe a positive measure of rectangles
Joshua Evan Greene, Andrew Lobb
Suppose that is a Jordan curve of diameter which encloses a region of area . We prove that there exists a subset of measure at least…
On the values taken by slice torus invariants
Peter Feller, Lukas Lewark, Andrew Lobb
We study the space of slice-torus invariants. In particular we characterize the set of values that slice-torus invariants may take on a given knot in terms of the stable smooth sli…
Squeezed Knots
Peter Feller, Lukas Lewark, Andrew Lobb
Squeezed knots are those knots that appear as slices of genus-minimizing oriented smooth cobordisms between positive and negative torus knots. We show that this class of knots is l…
Cyclic quadrilaterals and smooth Jordan curves
Joshua Evan Greene, Andrew Lobb
For every smooth Jordan curve and cyclic quadrilateral in the Euclidean plane, we show that there exists an orientation-preserving similarity taking the vertices of to…
The Rectangular Peg Problem
Joshua Evan Greene, Andrew Lobb
For every smooth Jordan curve and rectangle in the Euclidean plane, we show that there exists a rectangle similar to whose vertices lie on . The proof relies on Shev…
Threaded rings that swim in excitable media
Antonio Cincotti, Fabian Maucher, David Evans +6
Cardiac tissue and the Belousov-Zhabotinsky reaction provide two notable examples of excitable media that support scroll waves, in which a filament core is the source of spiral wav…
A refinement of Khovanov homology
Andrew Lobb, Liam Watson
We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus pri…
A counterexample to Batson's conjecture
Andrew Lobb
We show that the torus knot bounds a smooth Möbius band in the -ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth…
Almost positive links are strongly quasipositive
Peter Feller, Lukas Lewark, Andrew Lobb
We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow's in the (strong) positive. As a second m…
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…
The Dold-Whitney theorem and the Sato-Levine invariant
Andrew Lobb
We use the Dold-Whitney theorem classifying -bundles over a 4-complex to give a mod 4 obstruction to a 2-component link of trivial linking number being slice. It turns out t…
Upsilon-like concordance invariants from sl(n) knot cohomology
Lukas Lewark, Andrew Lobb
We construct smooth concordance invariants of knots which take the form of piecewise linear maps from [0,1] to R, one for each n greater than or equal to 2. These invariants arise…
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…
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…
On the functoriality of Khovanov-Floer theories
John A. Baldwin, Matthew Hedden, Andrew Lobb
We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting s…
Morse moves in flow categories
Dan Jones, Andrew Lobb, Dirk Schuetz
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified b…
An sl_n stable homotopy type for matched diagrams
Dan Jones, Andrew Lobb, Dirk Schuetz
There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to…
New Quantum Obstructions to Sliceness
Lukas Lewark, Andrew Lobb
It is well-known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a…
On spectral sequences from Khovanov homology
Andrew Lobb, Raphael Zentner
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a…
The quantum sl(n) graph invariant and a moduli space
Andrew Lobb, Raphael Zentner
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohts…
The Kanenobu knots and Khovanov-Rozansky homology
Andrew Lobb
Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first examp…
2-strand twisting and knots with identical quantum knot homologies
Andrew Lobb
Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homo…
A note on Gornik's perturbation of Khovanov-Rozansky homology
Andrew Lobb
We show that the information contained in the associated graded vector space to Gornik's version of Khovanov-Rozansky knot homology is equivalent to a single even integer s_n(K). F…