AN EXTENSION OF THE JONES POLYNOMIAL OF CLASSICAL KNOTS

We define a linear algebraic extension of the Jones polynomial of classical knots, and prove that certain key properties of the classical Jones polynomial are properties of the extension. This shows that these properties are linear algebraic in nature, not topological. We identify a topological property of the classical Jones polynomial, that is, a property of the classical Jones polynomial that the extension does not possess. We discuss ortho-projection matrices, ortho-projection graphs, and their Jones polynomials. We classify, up to isomorphism, the connected ortho-projection graphs with at most eight vertices, and show that each such isomorphism class corresponds to a prime alternating classical knot diagram. We give an example of a connected ortho-projection graph with nine vertices that does not correspond to such a diagram.