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.

The twisted torus knots lie on the standard genus 2 Heegaard surface for S³, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a previous paper by the author.

In the present paper, we will show that for any integer n > 0 there are infinitely many twisted torus knots with n-string essential tangle decompositions, and that all those knots have essential tori in the exteriors.

We show that a knot has a non-left-orderable surgery if the knot group admits a generalized Baumslag–Solitar relator and satisfies certain conditions on a longitude of the knot. As an application, it is shown that certain positively twisted torus knots admit non-left-orderable surgeries.

A twisted torus knot $T(p,q,r,s)$ is obtained from a torus knot $T(p,q)$ by twisting $r$ adjacent strands of $T(p,q)$ fully $s$ times. In this paper, we determine the parameters $p,q,r,s$ for which $T(p,q,r,s)$ is a torus knot with $s>0$.

A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ consecutive strands $s$ times fully twisted. We determine which twisted torus knots $T(p,q,p-kq,-1)$ are a torus knot.

A twisted torus knot $T(p,q,r,s)$ is a torus knot $T(p,q)$ with $r$ adjacent strands twisted fully $s$ times. In this paper, we determine the braid index of the knot $T(p,q,r,s)$ when the parameters $p,q,r$ satisfy $1<q<p<r\le p+q$. If the last parameter $s$ additionally satisfies $s=-1$, then we also determine the parameters $p,q,r$ for which $T(p,q,r,s)$ is a torus knot.

Twisted torus knots are torus knots with some full twists added along some number of adjacent strands. There are infinitely many known examples of twisted torus knots which are actually torus knots. We give eight more infinite families of such twisted torus knots with a single negative twist.

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.