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- 57-XX Manifolds and cell complexes
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In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating achiral knots:Let be a prime alternating achiral knot. Then there exists a minimal projection of in and an involution such that:1) reverses the orientation of ;2) ;3) ;4) has two fixed points on and hence reverses the orientation of .The purpose of this paper is to prove this statement.For the historical...
The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.
We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.
We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a -regular or -representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate the behavior of the non-acyclic Reidemeister torsion associated to a -bridge knot and -representations of its knot group.
We prove that the standard action of the mapping class group of a surface of sufficiently large genus on the unit tangent bundle is not homotopic to any smooth action.
The author gives an example showing that Thurston’s stability theorem cannot be generalized to non-oriented foliations.
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