Modifications of the bracket polynomial
DOI link for Modifications of the bracket polynomial
Modifications of the bracket polynomial book
The arrow polynomial is a decorated version of the f-polynomial introduced by Dye and Kauffman. The invariant incorporates extra information at each crossing and captures some information about the virtual crossings. The arrow polynomial is equivalent to the Miyazawa polynomial. The Miyazawa polynomial was introduced by Yasuyuki Miyazawa; the Miyazawa polynomial's definition begins with magnetic graph diagrams instead of virtual link diagrams. These two different approaches lead to equivalent invariants of virtual links. After the reduction, edges are defined by arrows and labeled with either 0 or 1. Consider a state with exactly one horizontal smoothing. Using the checkerboard framing, we see that a component of the state has either two arrows or zero arrows. Two connected arrows with positive weight result in a positive crossing. Two connected arrows with opposite weight result in a vertical smoothing. Next, construct a classical crossing at each virtual crossing by changing the edge labeled "1" to an over passing edge.