Intersecting Braids and Intersecting Knot Theory

An extension of the Artin Braid Group is considered, with the introduction of new operatores that generate double and triple intersections. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are also constructed via Markov traces defined on the intersecting braid algebra representations, and the extended Turaev representation is discussed as an example.