On Applications of Parity in Virtual Knot Theory

We investigate applications of parity in virtual knot theory and extend this philosophy to virtual links. This allows us to generalize previously known invariants - skein polynomials and biquandles. In the skein-theoretic direction we investigate graphical applications of parity for the bracket expansion of the Jones Polynomial as suggested by Manturov for virtual knots. Additionally we consider a similar extension to the Arrow Polynomial. We also show how this philosophy can be applied to the categorifications of both polynomials. In the process we show how known the new parity invariants produce information regarding minimal surface genus for virtual knots. Moreover, we provide calculators for these invariants and provide a list of the graphical polynomials for virtual knots with at most 4 real crossings. Similarly for biquandles, we use crossing parity to construct a generalization which we call Parity Biquandles. These structures include all biquandles as a standard example. Additionally, we find all Parity Biquandles arising from the Alexander Biquandle and Quaternionic Biquandles with integral coefficients. For a particular construction named the z-Parity Alexander Biquandle we show that the associated polynomial yields a lower bound on the number of odd crossings as well as the total number of real crossings and virtual crossings for the virtual knot. Moreover we extend this construction to links to produce a lower bound on the number of crossings between components of a virtual link.