Minimal generating sets of directed oriented Reidemeister moves

Abstract

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $8$ directed Polyak moves $\{ Ω{1a}^\uparrow, Ω{1a}^\downarrow, Ω{1b}^\uparrow, Ω{1b}^\downarrow, Ω{2a}^\uparrow, Ω{2a}^\downarrow, Ω{3a}^\uparrow, Ω{3a}^\downarrow \}$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least $2$ components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\Omega3$.