Outputs (25)

Squeezed knots (2024)
Journal Article
Feller, P., Lewark, L., & Lobb, A. (online). Squeezed knots. Quantum Topology, https://doi.org/10.4171/qt/187

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 large and discuss how to obstruct squeezedness. The most effective obst... Read More about Squeezed knots.

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, 234(3), 931–935. 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.

The rectangular peg problem (2021)
Journal Article
Greene, J., & Lobb, A. (2021). The rectangular peg problem. Annals of Mathematics, 194(2), 509-517. https://doi.org/10.4007/annals.2021.194.2.4

For every smooth Jordan curve and rectangle R in the Euclidean plane, we show that there exists a rectangle similar to R whose vertices lie on . The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smoo... Read More about The rectangular peg problem.

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.

A counterexample to Batson's conjecture (2019)
Journal Article
Lobb, A. (2019). A counterexample to Batson's conjecture. Mathematical Research Letters, 26(6), https://doi.org/10.4310/mrl.2019.v26.n6.a8

We give a counter-example to Batson’s conjecture on the non-orientable smooth slice genera of torus knots.

Threaded Rings that Swim in Excitable Media (2019)
Journal Article
Cincotti, A., Maucher, F., Evans, D., Chapin, B. M., Horner, K., Bromley, E., Lobb, A., Steed, J. W., & 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.