Remark 0.2. Note that if £* is a Heegaard splitting for M (i.e. bounding compression bodies) then there are many such curves /? on the boundary of a regular neighborhood of each cusp. If E* is not a Heegaard splitting and /3 is chosen to be a (0,1) element for some basis {a, (3} of the cuspital homology of the cusp in question, only (1, n) surgery on this cusp give manifolds in which Si corresponds to a Heegaard splitting (see [Sw]). Hence the corresponding Dehn surgery coordinate qi in Mq must have been (1, n). Remark 0.3. In accordance with current terminology one can view the Heegaard splittings where the cusps cores of the compression bodies are "vertical", and the case where a cusp can be isotoped onto the surface but is not a core as "horizontal" Heegaard splittings. In this case we can paraphrase Theorem 0.1(b) as saying that bounded genus irreducible Heegaard splittings of the manifolds in M' are either "vertical" or "horizontal". This is very similar to the situation in Heegaard splittings of Seifert fibered spaces (see [MS]).Let if be a knot in any 3-manifold M, in particular S 3 . A tunnel system for if is a collection of disjoint arcs ii,... , t s properly embedded in M-N(K) so that M-N(K U {Uii}) is a handlebody. Alternatively it is a collection of
Abstract. A bounded curvature path is a continuously differentiable piecewise C 2 path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature paths for points in the tangent bundle of the Euclidean plane. We show the existence of connected components of bounded curvature paths that do not correspond to those under (regular) homotopies obtaining the first results in the theory outside optimality. An application to robotics is presented. PreludeGiven a class of curves satisfying some constraints, understanding when there is a deformation connecting two curves in the class, where all the intermediate curves also are in the class, strongly relies on the defining conditions. When considering continuity any two plane curves (both closed or with different endpoints) are homotopic one into the other. Whitney in 1937 observed that under regular homotopies (homotopy through immersions) not always two planar closed curves lie in the same connected component [26]. In fact, there are as many regular homotopy classes of plane curves as integers. When considering curves with different endpoints the concept of homotopies through immersions do not lead to results different from those obtained when only continuity is considered. Dubins in 1957 introduced the concept of bounded curvature path when characterizing bounded curvature paths of minimal length [9] 1 .Let (x, X), (y, Y ) ∈ T R 2 be elements in the tangent bundle of the Euclidean plane. A planar bounded curvature path is a C 1 and piecewise C 2 path starting at x, finishing at y; with tangent vectors at these points X and Y respectively and having absolute curvature bounded by κ = 1 r > 0. Here r is the minimum allowed radius of curvature. The piecewise C 2 property comes naturally due to the nature of the length minimisers [9].A substantial part of the complexity of the theory of bounded curvature paths is described in the following observation. In general, length minimisers are used to establish a distance function between points in a manifold. This approach may not be considered for spaces of bounded curvature paths since in many cases the length variation between length minimisers of arbitrarily close endpoints or directions is discontinuous. Closely related is the fact that for most cases the length minimisers from (x, X) to (y, Y ) and from (y, Y ) to (x, X) have different length contradicting
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