Virtual Crossing Number and the Arrow Polynomial

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus. 1 The arrow polynomial We introduce the arrow polynomial, an invariant of oriented virtual knots and links that is equivalent to the simple extended bracket polnomial [6]. This invariant takes values in the ring Z[A,A, K1, K2, ...] where the Ki are an infinite set of independent commuting variables that also commute with the Laurent polynomial variable A.We give herein a very simple definition of this new invariant and investigate a number of its properties. This invariant was independently constructed by Miyazawa in [16] using a different definition. We do not make direct comparisons with the work of Miyazawa in this paper; such comparisons will be reserved for future work. From the arrow polynomial, we can obtain a lower bound on the virtual crossing number, determining in some cases whether a link is classical or