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Jones Polynomial

The Jones polynomial for virtual knots can be computed via the bracket polynomial in the same way as for classical knots, allowing for virtual crossings to exist in the closed loops, as described by Louis Kauffman in Virtual Knot Theory, Europ. J. Combinatorics (1999) 20, 663-691, arXiv:math.GT/9811028.

Cabled Jones Polynomials

Stronger virtual knot invariants can be formed by cabling the knot (adding one or more strands as shown below) and taking the Jones polynomial of the resulting link.

In order for this to work properly, twists, shown below, must be added in order to make the linking number of the two copies of the original knot equal to zero.

Behaviour on Mirror Images and Inverses

The cabled Jones polynomials are insensitive to inversion. Under both kinds of mirror images, q goes to q-1.