Virtual knots undetected by 1- and 2-strand bracket polynomials

Kishino's knot is not detected by the fundamental group or the bracket polynomial. However, we can show that Kishino's knot is not equivalent to the unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not detected by the 1-strand or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, these virtual knot diagrams are trivial as flats.