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On the values taken by slice torus invariants (2023)
Journal Article
FELLER, P., LEWARK, L., & LOBB, A. (2023). On the values taken by slice torus invariants. Mathematical Proceedings of the Cambridge Philosophical Society, 176(1), 55-63. https://doi.org/10.1017/s0305004123000403

We study the space of slice torus invariants. In particular we characterise the set of values that slice torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local Thom conj... Read More about On the values taken by slice torus invariants.

Cyclic quadrilaterals and smooth Jordan curves (2023)
Journal Article
Greene, J. E., & Lobb, A. (2023). Cyclic quadrilaterals and smooth Jordan curves. Inventiones Mathematicae, https://doi.org/10.1007/s00222-023-01212-6

For every smooth Jordan curve γ and cyclic quadrilateral Q in the Euclidean plane, we show that there exists an orientation-preserving similarity taking the vertices of Q to γ. The proof relies on the theorem of Polterovich and Viterbo that an embedd... Read More about Cyclic quadrilaterals and smooth Jordan curves.

A calculus for flow categories (2022)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2022). A calculus for flow categories. Advances in Mathematics, 409(Part B), Article 108665. https://doi.org/10.1016/j.aim.2022.108665

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 categories give rise to the same stable homotopy type of homological width... Read More about A calculus for flow categories.

Almost positive links are strongly quasipositive (2022)
Journal Article
Feller, P., Lewark, L., & Lobb, A. (2023). Almost positive links are strongly quasipositive. Mathematische Annalen, 385(1-2), 481-510. https://doi.org/10.1007/s00208-021-02328-x

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 main result, we give a simple and complete characterization of link dia... Read More about Almost positive links are strongly quasipositive.

A refinement of Khovanov homology (2021)
Journal Article
Lobb, A., & Watson, L. (2021). A refinement of Khovanov homology. Geometry & Topology, 25(4), 1861-1917. https://doi.org/10.2140/gt.2021.25.1861

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 primarily on strongly invertible knots and show, for instance, that this... Read More about A refinement of Khovanov homology.

On spectral sequences from Khovanov homology (2020)
Journal Article
Lobb, A., & Zentner, R. (2020). On spectral sequences from Khovanov homology. Algebraic & geometric topology, 20(2), 531-564. https://doi.org/10.2140/agt.2020.20.531

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realised as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1–ha... Read More about On spectral sequences from Khovanov homology.

Threaded Rings that Swim in Excitable Media (2019)
Journal Article
Cincotti, A., Maucher, F., Evans, D., Chapin, B. M., Horner, K., Bromley, E., …Sutcliffe, P. (2019). Threaded Rings that Swim in Excitable Media. Physical Review Letters, 123(25), Article 258102. https://doi.org/10.1103/physrevlett.123.258102

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 waves of excitation. Here we consider a novel topological configuration i... Read More about Threaded Rings that Swim in Excitable Media.

An sl(n) stable homotopy type for matched diagrams (2019)
Journal Article
Jones, D., Lobb, A., & Schuetz, D. (2019). An sl(n) stable homotopy type for matched diagrams. Advances in Mathematics, 356, Article 106816. https://doi.org/10.1016/j.aim.2019.106816

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 calculation. We lift this idea to the level of the Lipshitz-Sarkar sta... Read More about An sl(n) stable homotopy type for matched diagrams.

Upsilon-like concordance invariants from sl(n) knot cohomology (2019)
Journal Article
Lewark, L., & Lobb, A. (2019). Upsilon-like concordance invariants from sl(n) knot cohomology. Geometry & Topology, 23(2), 745-780. https://doi.org/10.2140/gt.2019.23.745

We construct smooth concordance invariants of knots K which take the form of piecewise linear maps Çn.K/W Œ0; 1 ! R for n 2. These invariants arise from sln knot cohomology. We verify some properties which are analogous to those of the invariant ‡ (... Read More about Upsilon-like concordance invariants from sl(n) knot cohomology.

Khovanov homotopy calculations using flow category calculus (2019)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2020). Khovanov homotopy calculations using flow category calculus. Experimental Mathematics, 29(4), 475-500. https://doi.org/10.1080/10586458.2018.1482805

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 two Steenrod squares. We develop a new algorithm which implements the f... Read More about Khovanov homotopy calculations using flow category calculus.

On the functoriality of Khovanov-Floer theories (2019)
Journal Article
Baldwin, J., Hedden, M., & Lobb, A. (2019). On the functoriality of Khovanov-Floer theories. Advances in Mathematics, 345, 1162-1205. https://doi.org/10.1016/j.aim.2019.01.026

We introduce the notion of a Khovanov–Floer theory. We prove that every page (after ) of the spectral sequence accompanying a Khovanov–Floer theory is a link invariant, and that an oriented link cobordism induces a map on each page which is an invari... Read More about On the functoriality of Khovanov-Floer theories.

Framed cobordism and flow category moves (2018)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2018). Framed cobordism and flow category moves. Algebraic & geometric topology, 18, 2821-2858. https://doi.org/10.2140/agt.2018.18.2821

Framed flow categories were introduced by Cohen, Jones and Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW complex with one cell for each object of the category. The idea is that th... Read More about Framed cobordism and flow category moves.

A Khovanov stable homotopy type for colored links (2017)
Journal Article
Lobb, A., Orson, P., & Schuetz, D. (2017). A Khovanov stable homotopy type for colored links. Algebraic & geometric topology, 17(2), 1261-1281. https://doi.org/10.2140/agt.2017.17.1261

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 coloring) of positive integer to each component of a link L, we define a s... Read More about A Khovanov stable homotopy type for colored links.

New Quantum Obstructions to Sliceness (2016)
Journal Article
Lewark, L., & Lobb, A. (2016). New Quantum Obstructions to Sliceness. Proceedings of the London Mathematical Society, 112(1), 81-114. https://doi.org/10.1112/plms/pdv068

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 ss. This gives a concordance homomorphism to the integers and a strong lower bound on... Read More about New Quantum Obstructions to Sliceness.

2–strand twisting and knots with identical quantum knot homologies (2014)
Journal Article
Lobb, A. (2014). 2–strand twisting and knots with identical quantum knot homologies. Geometry & Topology, 18(2), 873-895. https://doi.org/10.2140/gt.2014.18.873

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 homology and further algebraic structure which is then used to derive topo... Read More about 2–strand twisting and knots with identical quantum knot homologies.

The Kanenobu knots and Khovanov-Rozansky homology (2014)
Journal Article
Lobb, A. (2014). The Kanenobu knots and Khovanov-Rozansky homology. Proceedings of the American Mathematical Society, 142(4), 1447-1455. https://doi.org/10.1090/s0002-9939-2014-11863-6

Kanenobu has given infinite families of knots with the same HOMFLYPT polynomials. We show that these knots also have the same and HOMFLYPT homologies, thus giving the first example of an infinite family of knots indistinguishable by these invariants.... Read More about The Kanenobu knots and Khovanov-Rozansky homology.

The Quantum sl(N) Graph Invariant and a Moduli Space (2013)
Journal Article
Lobb, A., & Zentner, R. (2013). The Quantum sl(N) Graph Invariant and a Moduli Space. International Mathematics Research Notices, Advance Access, https://doi.org/10.1093/imrn/rns275

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, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thou... Read More about The Quantum sl(N) Graph Invariant and a Moduli Space.