Twisted knot polynomials: Inversion, mutation and concordance

The twisted Alexander polynomial of a knot is applied in three areas of knot theory: invertibility of knots, mutation, and concordance. Three examples are used to illustrate the utility of this invariant. First, a simple proof that the knot 8(17) is non-invertible is given. It is then proved that 8(17) is not even concordant to its inverse. Finally, the twisted polynomial is shown to distinguish the concordance class of the pretzel knot P(-3,5,7,2) from that of its positive mutant, P(5, - 3,7,2). This last example completes the solution to problem 1.53 of Kirby (1984, 1997) asking for a relation between mutation and concordance. (C) 1999 Elsevier Science Ltd. All rights reserved.