On the orientations of monotone knot diagrams

Abstract

An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given.