Bikei Cohomology and Counting Invariants

This paper gives a brief introduction into the fundaments of knot theory: introducing knot diagrams, knot invariants, and two techniques to determine whether or not two knots are ambient isotopic. After discussing the basics of knot theory an algebraic coloring of knots knows as a bikei is introduced. The algebraic structure as well as the various axioms that define a bikei are defined. Furthermore, an extension between the Alexander polynomial of a knot and the Alexander Bikei is made. The remainder of the paper is devoted to reintroducing a modified homology and cohomology theory for involutory biquandles known as bikei, first introduced in [18]. The bikei 2-cocycles can be utilized to enhance the counting invariant for unoriented knots and links as well as unoriented and non-orienteable knotted surfaces in R4