Narrow Results

In this work we present a direct connection between Garside normal forms in the braid group B₃ and their associated matrices in SL₂(ℕ) coming from the Nielsen–Thurston theory. As a consequence, we can give an upper and a lower bound for the dilatation factor of a rigid pseudo-Anosov three-strands braid with canonical length ℓ.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture $\mathbb{Z}$ is true for all the pretzel knots of the form $P(p,q,-r)$ where $p$, $q$ and $r$ are odd positive integers.

We present an one-to-one correspondence between (i) Pinto's golden tilings; (ii) smooth conjugacy classes of golden diffeomorphism of the circle that are fixed points of renormalization; (iii) smooth conjugacy classes of Anosov difeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugated to the Anosov automorphism G(x, y) = (x + y, x); (iv) solenoid functions. As we will explain, the solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences.