Topology Seminar: Dale Rolfsen

Classical knot groups, that is fundamental groups of knot complements in 3-space, are known to be torsion-free.  However, we show that for many knots, their groups contain generalized torsion: a nontrivial element such that some product of conjugates of that element equals the identity.  One example (the hyperbilic knot 5_2) was discovered with the aid of a Python program written by the USRA student Geoff Naylor.  Other examples include torus knots, algebraic knots in the sense of Milnor (arising from singularities of complex curves) and satellites of knots whose groups contain generalized torsion.  Although all knot groups are left-orderable, the existence of generalized torsion is an obstruction to their being bi-orderable.