Generalized torsion elements in the knot groups of twist knots (Q2796738)

An \(m\)-twist knot is given by linking together the ends of a \(2\)-braid with \(m\) full twists, according to a suitable convention depending on the sign of \(m\). If a non-trivial product of mutually conjugate elements in a group is the identity then the factors are called generalized torsion elements. Theorem: The knot group of any negative twist knot admits a generalized torsion element. This was so far known only for the \((-2)\)-twist knot \(5_2\). It is also known that in bi-orderable groups (allowing a strict total ordering invariant under multiplication on both sides), no generalized torsion elements exist, whence positive twist knots have no generalized torsion elements. The construction of a generalized torsion element of the Theorem is based on a presentation of the knot group derived from an appropriate Seifert surface.