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Kanenobu has given infinite families of knots with the same HOMFLYPT polynomials. We show that these knots also have the same sl(n)sl(n) and HOMFLYPT homologies, thus giving the first example of an infinite family of knots indistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.

Given a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

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 embedded Lagrangian torus in C2 has minimum Maslov number 2.

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 conjecture implies the existence of slice torus invariants without having to appeal to any explicit construction from a knot homology theory.

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 diagrams with quasipositive canonical surface (the surface produced by Seifert’s algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

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 obstructions appear to come from quantum knot invariants, notably including refinements of the Rasmussen invariant due to Lipshitz–Sarkar and Sarkar–Scaduto–Stoffregen involving stable cohomology operations on Khovanov homology.

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 by the Morse moves. These include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index, and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction associates a stable homotopy type with a framed flow category whose cohomology is designed to recover the corresponding Morse or Floer cohomology. We obtain analogues of the Whitney trick and of handle cancellation for framed flow categories: in this new setting, these are moves that can be performed to simplify a framed flow category without changing the associated stable homotopy type. These moves often enable one to compute by hand the stable homotopy type associated with a framed flow category. We apply this in the setting of the Lipshitz-Sarkar stable homotopy type (corresponding to Khovanov cohomology) and the stable homotopy type of a matched diagram due to the authors (corresponding to 𝔰𝔩n Khovanov-Rozansky cohomology).

Ribbon concordance gives a partial order on knot types, and applying a knot homology functor to a ribbon concordance gives an inclusion of the homologies. The question of the existence of global ribbon minima in each concordance class is a generalization of the slice-ribbon conjecture, which asserts that the unknot is the global minimum in its class. We show that the (reduced rational) Khovanov homology of the (4,5) torus knot is a summand in the Khovanov homology of any knot in its concordance class.

Forms of non-random copying error provide sources of inherited variation yet their effects on cultural evolutionary dynamics are poorly understood. Focusing on variation in granny and reef knot forms, we present a mathematical model that specifies how these variant frequencies are affected by non-linear interactions between copying fidelity, mirroring, handedness and repetition biases. Experiments on adult humans allowed these effects to be estimated using approximate Bayesian computation and the model is iterated to explain the prevalence of granny over reef knots in the wild. Our study system also serves to show conditions under which copying fidelity drives heterogeneity in cultural variants at equilibrium, and that interaction between unbiased forms of copying error can skew cultural variation.