TWISTED KNOT POLYNOMIALS: INVERSION, MUTATION AND CONCORDANCE

Abstract

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.