Narrow Results

We determine the knot types of twisted torus knots T(p, q;r, s) when q divides r. We also determine their tunnel numbers.

We study the essential surfaces in the exterior of a cable knot to compute the representativity and waist of most cable knots. Our computation answers Ozawa’s question [5] about the relationship between the representativity and the waist of a knot in the negative.

The twisted torus knot $T(p,q,r,s)$ is a knot obtained from the torus knot $T(p,q)$ by twisting $r$ adjacent stands fully $s$ times. If $r=p$ or $r$ is a multiple of $q$, then $T(p,q,r,s)$ is known to be a cable of a nontrivial knot or a torus knot. Assuming that $r\ne p$ and $r$ is not a multiple of $q$, we determine the parameters $p,q,r,s$ for which the positively twisted torus knot $T(p,q,r,s)$ is a cable knot.

Consider the knots obtained from torus knots by adding a negative full twist along two adjacent strands. Among these knots, we determine which are cable knots or torus knots.