Unoriented links and the Jones polynomial

Abstract

The Jones polynomial is an invariant of oriented links with $n\ge 1$ components. When $n=1$, the choice of orientation does not affect the polynomial, but for $n>1$, changing orientations of some (but not all) components can change the polynomial. Here we define a version of the Jones polynomial that is an invariant of unoriented links; i.e., changing orientation of any sublink does not affect the polynomial. This invariant shares some, but not all, of the properties of the Jones polynomial.

The construction of this invariant also reveals new information about the original Jones polynomial. Specifically, we show that the Jones polynomial of a knot is never the product of a nontrivial monomial with another Jones polynomial.