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A proof of Tait’s Conjecture on prime alternating - achiral knots

Nicola Ermotti, Cam Van Quach Hongler, Claude Weber (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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 K be a prime alternating - achiral knot. Then there exists a minimal projection Π of K in S 2 ⊂ S 3 and an involution ϕ : S 3 → S 3 such that:1) ϕ reverses the orientation of S 3 ;2) ϕ ( S 2 ) = S 2 ;3) ϕ ( Π ) = Π ;4) ϕ has two fixed points on Π and hence reverses the orientation of K .The purpose of this paper is to prove this statement.For the historical...

A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization

Robert Feßler (1995)

Annales Polonici Mathematici

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.

A regularity lemma for functions of several variables.

Jean L. Journé (1988)

Revista Matemática Iberoamericana

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.

A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion

Yoshikazu Yamaguchi (2008)

Annales de l’institut Fourier

We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a λ -regular SU ( 2 ) or SL ( 2 , ℂ ) -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 2 -bridge knot and SU ( 2 ) -representations of its knot group.

A remark on Thurston's stability theorem

Richard Sacksteder (1975)

Annales de l'institut Fourier

The author gives an example showing that Thurston’s stability theorem cannot be generalized to non-oriented foliations.

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