Alexander polynomial for torus knots via Burau matrices for periodic braids

Abstract

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This thesis provides a characterization of the reduced Burau matrices for braids of the form (σ1σ2 · · · σn−1) d , with gcd(n, d) = 1, n, d ≥ 2, and exposes its relationship with the Alexander polynomial for (n, d)-torus knot by using Markov functions theory. In addition, a similar characterization for a particular case of periodic braids is provided, whose closures is the mirror of a (n, d)-torus knot.

Esta tesis provee una caraterizaci´on de las matrices reducidas de Burau para trenzas de la forma (σ1σ2 · · · σn−1) d , con mcd(n, d) = 1, n, d ≥ 2, y expone su relaci´on con el polinomio de Alexander para nudos toroidales, usando la teoria de funciones de Markov. En adici´on, proporcionamos una caracterizaci´on similar para un caso particular de trenzas periodicas cuya clausura es el espejo de un (n, d)-nudo toroidal.