Minimal generating sets of directed oriented Reidemeister moves

Abstract

Polyak proved that the set [Formula: see text] is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining [Formula: see text] different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves [Formula: see text] is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a [Formula: see text]-generating set for a link [Formula: see text]. The same set is proven to be a minimal [Formula: see text]-generating set for any link [Formula: see text] with at least two components. Finally, we discuss knot diagram invariants arising in the study of [Formula: see text]-generating sets for an arbitrary knot [Formula: see text], emphasizing the distinction between ascending and descending moves of type [Formula: see text].