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 stable homotopy type and use it to make new computations.Similarly, there exists a simplified Khovanov-Rozansky sln complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n ≥ 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n ≥ 3 the cohomology of the stable homotopy type agrees with the sln Khovanov-Rozansky cohomology of the underlying knot.We make some consistency checks of this sln stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP 2 in the sl 3 type for some diagram, and show that the sl 4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces. AL and DS were both supported by EPSRC grant EP/M000389/1, DJ was supported by an EPSRC graduate studentship.

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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 moves 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 to 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 to 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).

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There is an increasing demand for piloted autonomous underwater vehicles (AUVs) to operate in polar ice conditions. At present, AUVs are deployed from ships and directly human-piloted in these regions, entailing a high carbon cost and limiting the scope of operations. A key requirement for long-term autonomous missions is a long-range route planning capability that is aware of the changing ice conditions. In this paper we address the problem of automating long-range route-planning for AUVs operating in the Southern Ocean. We present the route-planning method and results showing that efficient, ice-avoiding, long-distance traverses can be planned. This paper is under review 1 .