Twisted torus knots that are unknotted (Q2927902)

In this low-dimensional topology article, the author characterizes unknotted twisted torus knots.NEWLINENEWLINEThe \textit{twisted torus knot} \(T(p,q,r,s)\) is obtained by introducing \(s\) full-twists on \(r\) adjacent strands of the torus knot \(T(p,q)\). Here, \(p, q, r, s\) are integers with \(p\) and \(q\) coprime, \(1 \leq q < p\), \(2 \leq r\leq p+q\), and \(s\neq 0\). The main result characterizes the quadruples \((p,q,r,s)\) for which \(T(p,q,r,s)\) is unknotted in terms of seven mutually disjoint families. As a consequence, the author finds families of torus knots that can be unknotted by performing \(\frac{1}{s}\)-surgery along an unknot for an integer \(s\). It turns out that this includes the torus knots \(T(f_{n+1},f_{n-1})\), where \(f_{n}\) denotes the Fibonacci sequence.NEWLINENEWLINEThe proof of the main result is structured as follows. If \(p\) equals \(r\) or \(q\) divides \(r\), then \(T(p,q,r,s)\) is a torus knot or a cable of a torus knot for which the author quickly decides whether or not it is unknotted. Otherwise, the author uses the following point of view. The twisted torus knot \(T(p,q,r,s)\) is obtained from \(T(p,q)\) by \(\frac{1}{s}\)-surgery along an unknotted curve \(C\). The main part of the argument (and the paper) consists of showing that the seven families provided in the main theorem are the only twisted torus knots that are unknotted. This is done by showing that \(T(p,q)\cup C\) is a hyperbolic link, which is used to restrict the values of \(p,q,r,\) and \(s\) for unknotted \(T(p,q,r,s)\). Besides classical \(3\)-manifold results, the latter involves combinatorial group theory observations about the free group on two generators and explicit isotopies between twisted torus knots.