On The Number of Components of Twisted Torus Links

Twisted torus links T(p, q; r, s) generalize torus links by introducing s additional twists on r adjacent strands of the torus link T(p, q). It is well known that the number of components of a torus link T(p, q) is given by the greatest common divisor of p and q. However, the number of components of a twisted torus link cannot be determined as straightforwardly from its parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links from their parameters. As a result, we show that the number of components of a twisted torus link T(p, q; r, s) is a multiple of gcd(p, q, r, s), and in particular, T(p, q; r, s) is a knot only if gcd(p, q, r, s) = 1. We also use our algorithm to prove several conjectures related to the number of components in twisted torus links.