We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S 3 . Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S 3 .05C10, 57M25
Abstract. We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.
We prove that for 2-bridge knots and 3-bridge knots in thin position the double branched cover inherits a manifold decomposition in thin position. We also argue that one should not expect this sort of correspondence to hold in general.
PreliminariesThin position for knots was first defined by D. Gabai in [2]. He used this notion to prove Property R for knots. A few years later M. Scharlemann and A. Thomspon developed a notion of thin position for 3-manifolds. Both of these notions have become vital tools in many geometric arguments. We here investigate the similarities and differences between the two notions. Definition 1.1. Suppose L is a submanifold of M. We will denote an open regular neighborhood of L in M by η(L, M) or simply by η(L), if the ambient manifold is clear from the context. Similarly, we will denote a closed regular neighborhood of L in M by N(L, M) or simply by N(L).We define thin position for knots as in [15]: Definition 1.2. Let k be a knot in the 3-sphere. The complement of k is S 3 − η(k). Definition 1.3. A meridional planar surface in the complement of k is a planar surface properly embedded in the knot complement whose boundary components are meridians. Definition 1.4. A boundary parallel annulus with meridional boundary components is a trivial meridional planar surface.
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