Properties
The Jones polynomial is characterized by the fact that it takes the value 1 on any diagram of the unknot and satisfies the following skein relation:
where, and are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:
The definition of the Jones polynomial by the bracket makes it simple to show that for a knot, the Jones polynomial of its mirror image is given by substitution of for in . Thus, an amphichiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations.
Read more about this topic: Jones Polynomial
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