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 817 is non-invertible is given. It is then proved that 817 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.