An Extended Bracket Polynomial for Virtual Knots and Links

Abstract

This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by < > for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the polynomial takes values in the module generated by these reduced graphs over the ring Q[A,A^{-1}]. The paper is relatively self-contained, with background information about virtual knots and long virtual knots. We give numerous examples applying the extended bracket, including a new proof of the non-triviality of the Kishino diagram and the flat Kishino diagram and non-classicality of single crossing virtualizations. The paper has a section on the estimation of virtual crossing number using the extended bracket state sum. Examples are given of virtual knots with arbitrary minimal embedding genus and arbitrarily high positive difference between the virtual crossing number and the minimal embedding genus. A simplification of < > is introduced and denoted by A[K]. This simplified extended bracket, the arrow polynomial, is a polynomial in an infinite set of variables. It is quite strong (detecting the flat Kishino diagram for example) and easily computable. The paper contains a description of a computer algorithm for A[K] and uses the arrow polynomial, in conjunction with the extended bracket polynomial to determine the minimum genus surfaces on which some virtual knots can be represented.